I am having trouble understanding why the push c4 here is incorrect for Black.
After 9. Qc2 b5 10. a4 a6 11. axb5 axb5 12. Rxa8 can be recaptured with the Queen and the pawn chain is stable.
What is wrong then with this push?
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First, although it is hard to disagree with the computer there, it is interesting that 8...c4 scores quite well for black in the Mega 2019 database (only a 43.8% for white).
The first obvious thing that comes to mind is that anytime you change the base of attack on the pawn chain like that it does two related things (right now the base of attack is at d4 and black can attack it with c5...after c4 the new base is c3, and black will need to attack it with b5-b4): Typically when you push, it often makes the later break with b5-b4 stronger, but of course it does not come without a price...it is much slower so you may never get to play that break, or it might be ineffectual by then, especially on the q-side since with similar advances on the k-side threats to the king cannot be ignored. A very common example of this is in the King's Indian Classical Variation, when black plays f5-f4. It is slower, but there are countless examples showing just how dangerous black's attack may become if white's counterplay is not very active. Sometimes, white's q-side counterplay can even be ignored, even if it costs a rook or other significant material due to unstoppable mating threats.
In this position, specifically, you are already behind in development, so slowing your counterplay on the q-side may well be fatal. Right now, until you are better developed, and can see if white might somehow not continue to develop effectively, it is too early to change the attack point of white's pawn chain. If you catch up in development, and see that white cannot attack your castled king with impunity, then maybe later you can opt for c4.
A better plan is to play Ne7-c6 (it may also go to f5) and pressure d4. Other common moves in that plan will be Be7, 0-0, and possibly cd and Qb6 with Rc8 to follow.
In the line that you gave, after 8...c4; 9 Qc2 b5, instead of immediately taking on black's q-side, I think white should just continue developing to make your lack of development more pronounced. I even suggest the very "sophisticated" 10.Bg5! Be7; 11. Be3! when black has to figure out how to finish developing his k-side since 11....Nh6 is just bad there as is 11....f6. I think black is in trouble in the long run.
c4 is the usual beginner's mistake of releasing the tension unnecessarily. With the pawn on c5 you exert some pressure on d4 which can be increased with moves like Ne7-c6. When the pawn moves to c4 that pressure disappears.
If you played Ne7 instead then taking the c5 pawn is bad for white. You are not going to retake immediately but instead play Nc6. This gives white the choice between allowing you to recapture on c5 with the bishop with gain of tempo or allowing you to take the more important e5 pawn. Instead of taking on c5 white will want to develop his queenside pieces but this is awkward in the current position. His dark squared bishop is problematic.
After 9. Qc2 b5 10. a4 a6 11. axb5 axb5 12. Rxa8
This is "hope" chess. You "hope" your opponent is going to make some bad moves and then your bad move magically becomes a good move. If you want to improve then you have to stop doing this.
Your opponent is not going to play Qc2 because if the c file opens then he wants a rook on that file not the queen. Qe2 is better. He is also not going to play 10. a4. He is going to play 10. b3, immediately attacking the front of your pawn chain.
Furthermore you are a long way behind in development with your king still stuck in the middle. You should not be wasting time on further pawn moves, particularly ones which threaten to open the position leaving your king exposed to attack. Ne7 is more to the point, developing a piece and getting you one move closer to being able to castle.