[title "16.c4 without Nd7"]
 [fen ""]
 [StartPly "34"]

 1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Ba4 Nf6 5. O-O Be7 6. Re1 b5 7. Bb3 d6 8. c3 O-O 9. h3 Bb7 10. d4 Re8 11. Nbd2 Bf8 12. a4 h6 13. Bc2 exd4 14. cxd4 Nb4 15. Bb1 c5 16. d5 c4 17. Nd4 {b5 pawn is attacked twice now} Nd7 {this is what happened in the game, but Nd7 here is too late} (17...Qb6 {an attempt to directly defend the pawn, but the queen ends up being completely misplaced as we'll see and too far away from the kingside.} 18. N2f3 {defending the d4 knight and preparing Be3} g6 19. Be3 Qc7 {side-stepping the discovery} (19...bxa4 {just for the sake of discussion, let's see how black loses by force if they do not side step the discovered threat and take on a4 instead} 20. Ne6 Qb5 21. Nxf8 Rxf8 22. Qd4 {and white's kingside attack is too much to parry as the king on g8 is defenceless}) 20. axb5 {winning the b5 pawn if black recaptures, so only chance is to take on e4 instead} Nxe4 21. Ne6 {and white's nearly winning already as black's position is collapsing after fxe6 Bxe4. Ne6 is a thematic move in these structures with the pawn on d5.} ) 18. axb5 axb5 19. Rxa8 Bxa8 20. Nxb5 Qb8 21. Na3 {c4 is falling next, as happened in the game} Nd3 22. Re3 Nxc1 23. Qxc1 g6 24. Naxc4 {rest of the game is not included here}

 [title "16.c4 without Nd7"]
 [fen ""]
 [StartPly "33"]

 1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Ba4 Nf6 5. O-O Be7 6. Re1 b5 7. Bb3 d6 8. c3 O-O 9. h3 Bb7 10. d4 Re8 11. Nbd2 Bf8 12. a4 h6 13. Bc2 exd4 14. cxd4 Nb4 15. Bb1 c5 16. d5 c4 17. Nd4 {b5 pawn is attacked twice now} Nd7 {this is what happened in the game, but Nd7 here is too late} (17...Qb6 {an attempt to directly defend the pawn, but the queen ends up being completely misplaced as we'll see and too far away from the kingside.} 18. N2f3 {defending the d4 knight and preparing Be3} g6 19. Be3 Qc7 {side-stepping the discovery} (19...bxa4 {just for the sake of discussion, let's see how black loses by force if they do not side step the discovered threat and take on a4 instead} 20. Ne6 Qb5 21. Nxf8 Rxf8 22. Qd4 {and white's kingside attack is too much to parry as the king on g8 is defenceless}) 20. axb5 {winning the b5 pawn if black recaptures, so only chance is to take on e4 instead} Nxe4 21. Ne6 {and white's nearly winning already as black's position is collapsing after fxe6 Bxe4. Ne6 is a thematic move in these structures with the pawn on d5.} ) 18. axb5 axb5 19. Rxa8 Bxa8 20. Nxb5 Qb8 21. Na3 {c4 is falling next, as happened in the game} Nd3 22. Re3 Nxc1 23. Qxc1 g6 24. Naxc4 {rest of the game is not included here}

That said, and although your hunch about the early c4 is correct, I'm afraid there's no simple answer that immediately clarifies as to why the immediate c4 is bad and why Nd7 is so crucial, since the Ruy Lopez and the Zaitsev are highly theoretical openings and thus, all about nuances and concrete assessments.

Bit of context first: the game started out as a classical Ruy Lopez which then transitioned into the Zaitsev variation. Then in particular after 11...Bf8, Firouzja opted for the older system with 12.a4 which is a highly rich variation in theory dating specially back to the Kasparov-Karpov games (see e.g. 1990 games). Instead of 12.a4, nowadays the more go-to and modern option is 12.d5. So, seeing 12.a4 might have come as a bit of a surprise.

However, Kovalev played the immediate c4 advance without the inclusion of Nd7, this was the main inaccuracy! The problem is black's losing a pawn in the all ensuing lines without any dynamical or positional compensation for it. Let's see some of these lines in the annotated diagram below:

 [title "Correct move order: Nd7 before c4, actually setting up Nd3"]
 [fen "r2qrbk1/1b3pp1/p2p1n1p/1ppP4/Pn2P3/5N1P/1P1N1PP1/RBBQR1K1 b - - 0 16"]
 [startflipped "0"]

 1...Nd7 {black entertains both options of f5 and c4 after Nd7} 2. Ra3 {These are nuanced moves which were specially polished in Kasparov-Karpov games of 90's. Ra3 is a multi-facetted move, moving the rook to a protected square so that axb5 axb5 lines does not force white to take rooks at once, moreover activates the rook along the 3rd row both additional coverage of d3 square and potential rook swings to kingside} c4 3. axb5 axb5 4. Nd4 Rxa3 5. bxa3 Nd3 {black appears to be losing a pawn nonetheless, however as demonstrated by Morozevich among others, black is in fact able to maintain the awkwardly advanced d3 pawn with Nc5 which gives black sufficient dynamic compensation for the pawn} 6. Bxd3 cxd3 7. Re3 Nc5 {and black is well on time with Nc5 as opposed to what transpired in the game where black delayed Nd7 to after c4!} 8. Nxb5 {not white's best choice} (8. Bb2 {white's best choice here is not to take the pawn as that gives black too much counter-play.}) Qa5 9. Nd4 Qc3 {and Qa1 threats are constantly in the air. Additionally, the knight on d2 is bound to the defense of the e4 pawn. Black is clearly better.}

It would go beyond the scope of this post to exhaust all possible subvaruations both with and without Nd7, however I hope the above discussions and lines exemplify the importance of the inclusion of Nd7 (and the playability of the ensuing lines) in contrast to the inaccurate early c4 without Nd7 as happened in the game.

That said, and although your hunch about the early c4 is correct, I'm afraid there's no simple answer that immediately explains why the immediate c4 is bad and why Nd7 is so crucial, since the Ruy Lopez and the Zaitsev are highly theoretical openings and thus, all about nuances and concrete assessments.

Bit of context first: the game started out as a classical Ruy Lopez which then transitioned into the Zaitsev variation. Then in particular after 11...Bf8, Firouzja opted for the older system with 12.a4 which is a variation rich in theory dating specially back to the Kasparov-Karpov games (see e.g. 1990 games). Instead of 12.a4, nowadays the more go-to and modern option is 12.d5. So, seeing 12.a4 might have come as a bit of a surprise.

However, Kovalev played the immediate c4 advance without the inclusion of Nd7, this was the main inaccuracy! The problem is black's losing a pawn in the all ensuing lines without any dynamic or positional compensation for it. Let's see some of these lines in the annotated diagram below:

 [title "Correct move order: Nd7 before c4, actually setting up Nd3"]
 [fen "r2qrbk1/1b3pp1/p2p1n1p/1ppP4/Pn2P3/5N1P/1P1N1PP1/RBBQR1K1 b - - 0 16"]
 [startflipped "0"]

 1...Nd7 {black entertains both options of f5 and c4 after Nd7} 2. Ra3 {These are nuanced moves which were specially polished in Kasparov-Karpov games of 90's. Ra3 is a multi-faceted move, moving the rook to a protected square so that axb5 axb5 lines do not force white to exchange rooks at once, moreover activates the rook along the 3rd row both additional coverage of d3 square and potential rook swings to kingside} c4 3. axb5 axb5 4. Nd4 Rxa3 5. bxa3 Nd3 {black appears to be losing a pawn nonetheless, however as demonstrated by Morozevich among others, black is in fact able to maintain the awkwardly advanced d3 pawn with Nc5 which gives black sufficient dynamic compensation for the pawn} 6. Bxd3 cxd3 7. Re3 Nc5 {and black is well on time with Nc5 as opposed to what transpired in the game where black delayed Nd7 to after c4!} 8. Nxb5 {not white's best choice} (8. Bb2 {white's best choice here is not to take the pawn as that gives black too much counter-play.}) Qa5 9. Nd4 Qc3 {and Qa1 threats are constantly in the air. Additionally, the knight on d2 is bound to the defense of the e4 pawn. Black is clearly better.}

It would go beyond the scope of this post to exhaust all possible sub variations both with and without Nd7, however I hope the above discussion and lines clarify the importance of the inclusion of Nd7 (and the playability of the ensuing lines) in contrast to the inaccurate early c4 without Nd7 as happened in the game.