Peeled Rubik's Cube

Question

Help! A [toddler / pet / washing machine] has peeled some of the stickers off my old off-brand Rubik's Cube. Can you still solve it and show me the result?

Answer 1

First, we tabulate all the adjacencies:

Then suppose that White and Orange are opposite:

This leaves G and R adjacent to everything: a contradiction.
So O is adjacent to W:

Now we need to look in a little more detail. We'll start with the corner pieces. I'm going to rotate the center pieces around to make it so that there are fewer moves. We also need to remember the parity rules for a rubik's cube:

Parity rules are: (1) Even number of swaps, (2) even number of edge pieces twisted, (3) clockwise twists of corners = 0 modulo 3.
Ignore the edge pieces for now. The top net is what we're given (plus a few center pieces I've colored for convenience). The table on the right is the transformation of the corners. And the bottom net is the result.
Also, I'm indexing the corner pieces as follows:


In this scenario, corner 8 must be piece 6 (for the G and W). If Red and White goes to position 7, then Yellow and White will have to go to position 5. This establishes the colors of the faces. It also makes a contradiction, because the Blue and Red must go to position 4 and the Blue and Green must go to position 4 (shown in position 3 in the bottom net).
So this is impossible.
Similarly, if the Red face is opposite the Green face, there is a clash.
So, with the corners completed, the position is as follows:

The sum of the rotations is 6, so that's good.
The number of swaps is odd, which means that the edge pieces will need an odd number of swaps also.

So now we need to work on the edges!

The indexing we'll use for the edges is:

Then we can easily slot the various 2 colored edges into their correct places:

You'll notice that there are 6 edges to be allocated, each containing one of the colors. And there are two colors missing from each face. So guess that Edge 1 is Orange and Red, and chase from there:

Checking this solution on an on-line solve demonstrates that it is impossible.
So we try the other option: Edge 1 is O & W. Then we get this:

This one can be solved with the online solver, so it is the correct answer.

Hence the final answer is the last diagram above. Or, with the OP's original diagram:

Answer 2

The peeled cube looks like this in its solved state: