If you imagine a polynomial L(z) that's zero at all the missing numbers, you can expand the coefficients out. For example, with 2 missing numbers (x,y), you have:

   L(z) = z^2 - (x+y)z + xy.

Then you solve for the roots of L, either using your finite field's variant of the quadratic formula, or e.g. just by trying everything in the field.

* But wait, this doesn't actually work! *

Over fields of small characteristic, such as F_2^m, you need to modify the approach and use different powers. For example, in the equations above, I divided by 2. But over F_2^m in the example shown above, you cannot divide by 2, since 2=0. In fact, you cannot solve for (x,y) at all with only x+y and x^2 + y^2, because

  (x+y)^2   =   x^2 + y^2 + 2xy   =   x^2 + y^2 + 0xy (since 2=0)   =   x^2 + y^2