At this point, I believe white is completely lost, and it seems that the engine agrees. I think the basic idea is that the material deficit is too much and the black king isn't in any real danger as the bishop on c4 can come into g7 and block any checks/mating ideas. While the white king is in the open with active black pieces around it.
Firstly if you take the rook (Qxa8), then Qc4. You can block any checks with the bishop and create a discovery check, which will lead to a checkmate.
You can't really protect the c-pawn well and if you do, it will be too slow as black has very quick ways to infiltrate with the queen. And for similar reasons, it is also not feasible to play defence, as you have too many weaknesses to cover
Trading queens also doesn't get you anywhere, although it is recommended by the engine since black's queen is more dangerous than white's. In that case, if you take back with the rook (Rxd7) then Bg7 again and black's rook is always going to keep an eye on the pawn. If pawn takes, then Bf6 and the square is always covered enough times. Not to mention that black can always sack a piece for the pawn and will still be up a minor piece.
Having said all that, although the game is lost for white in all situations, I think your best practical bet would be to take the rook on a8 and sort of hope that black is not able to find the checkmate. As counterplay, you can try to push your past pawn or get your rook on offence with a check.
Hope this helps!