Randomized Algorithm in $O(d)$ for Solving Unknown Degree $d$ Polynomial Function Using an Erroneous Oracle
Some hints:

Could you solve the problem if the function was of the form $f : GF_p \to GF_p$, i.e., a univariate polynomial?  I suggest solving that special case first.  The below are some hints on how to do that:

If the error was zero, would you know what to do?  Could you devise a strategy that would always work in that situation?  How many queries would your strategy need?

Now suppose you applied your strategy, in the situation where there could be errors (but with the error probability bounded as you've described).  What's the probability that your strategy gives the wrong answer?  What's the probability that your strategy gives the right answer?

You might find it useful to use a union bound.

Can you think of any way to check whether a possible solution is correct?

Finally, once you can solve the univariate case, can you think of any way to repeat that procedure multiple times to handle the situation of a polynomial in two variables?