There's quite a lot of chaff in the question, but as far as I can tell, we're supposed to do retrograde things to the situation to find out which piece(s) have been removed from the board.
Part 1: Roll call
So, let's start with white: we can see all the pieces except the dark square bishop. Its fate is clear: black has captured it earlier on. (It cannot have moved from its starting square at C1, yet the white king must have moved through that square to get to its current position.)
Since black has only captured at C1, all the black pawns must therefore still be on their starting files. In particular, the g- and h-pawns are still there, so black hasn't promoted any pawns.
Next, let's count the rest of the black (non-pawn) pieces. At least three of them have been taken by the white pawns (1 by the g pawn, 2 by the h pawn), and since we can see two rooks, a bishop and a knight, that actually accounts for all the black pieces.
Part 2: Previous moves
It clearly cannot be white's turn: no black piece has a square it could have come from.
If it's now black's turn, then white's previous move cannot have been anywhere at the top right: the f7 pawn cannot have come from e6, because while it could have gotten to the e file by taking the e pawn, getting back would require an extra captured black piece. The rook doesn't have any squares it could have come from, and the knight cannot have moved into its current position either: the black king would have been in check during white's turn otherwise. (Because the knight move cannot have captured anything.)
This means black's previous move was not made with any of the pieces still on board. It also cannot be by another piece that got captured since, because black's missing pieces were taken by white pawns, so the previous white move would have needed to be
gxf3, which is impossible, because white's light square bishop couldn't have left its starting square with g2 still in place.
The position as shown is illegal
But since we are given the possibility that there are some pieces missing from the board, there's still hope.
Part 3: Untangling the clump
The tangle at the top right is super tight. I cannot find any other way to reach it except that the black king must be the very last piece to step in, and that can only have occurred instantly after it was checked by the knight at h5.
But this this is utterly ridiculous. It means that at the time the tangle was finalized, white's g pawn was still on g2. And therefore white's light square bishop was on f1! And this means the white pieces at the home rank cannot yet have changed their order at this time.
FEN: q5bn/5PrR/5pr1/5Np1/6kp/8/PPPPPPP1/R2QKBN1 w - - 0 1
In the picture above, I've returned the black piece to be captured on f3 to the board (the queen at A8). The other white knight is on its home square, only to keep things simple. (We're going to want to dodge with it later, but we'll do it after the bishop is out)
From here, we can now finalize the tangle with
1. Nh6+ Kh5 2. Rb1 Qf3 3.gxf3, but then we are out of a black move. Or about a billion of them, really. Let's count:
bishop to A8: 4 moves
King to f2 and then A1: 8 moves
Knight to dodge the heavy pieces: 2 moves
Queen to h1 and back: 2 moves
Rook to g1 and back: 2 moves
That's quite a few moves. So many, actually, (> 16) that we must return all the black pawns to the board. We cannot return all of them in their starting squares though, because the C8 bishop needs a way out.
Looks like we're not unique yet, though. From here, the given position can be reached with the sequence
b6 2.Bh3 c6 3.Bc8 a6 4.Bb7 a5 5.Ba8 b5 6.Nh3 c5 7.Kf1 d5 8.Kg2 e6 9.Qh1 e5 10.Rg1 a4 11.Kf1 b4 12.Ke1 c4 13.Kd1 d4 14.Kc1 e4 15.Kb1 a3 16.Ka1 b3 17.Rb1 c3 18.Qd1 d3 19.Ng1, and there's still ambiguity in the pawn positions:
Maybe there are some hidden complications in the earlier phases of clump creation?
Ah, yes. One of white's h-pawn's captures must be the dark square bishop. That capture must have happened either behind the black g-pawn (impossible, there are no suitable dark squares), on f7 (wrong coloured square), or before the g pawn moved. That means that black's e-pawn must also have moved before the clump was formed, and finally we have a unique solution:
The missing pieces are
The five black pawns, all pushed as far forward as they can go,
and it's black's turn to move from this position: