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Not an answer, but it doesn't fit in a comment box and relevant, I think. With:

def test1():
    count = 0
    for a in k:
        for b in k:
            if not f.subs(x=a, y=b):
                count += 1
    return count

def test2():
    count = 0
    for a in k:
        for b in k:
            if not f(a, b):
                count += 1
    return count

def test3():
    count = 0
    for a in k:
        for b in k:
            if not (b^3+b+4*a):
                count += 1
    return count

we get

sage: timeit('test1()')
25 loops, best of 3: 24.6 ms per loop
sage: timeit('test2()')
5 loops, best of 3: 64.1 ms per loop
sage: timeit('test3()')
125 loops, best of 3: 2.73 ms per loop

so, surprisingly, subs is quite a bit faster than "polynomial evaluation", and unsurprisingly writing out the expression explicitly is of course even faster than either.

The subs method gives back a result in a ring that's a little too large for the purposes, though (this is as documented):

sage: parent(f.subs(x=k(1),y=k(1))) 
Multivariate Polynomial Ring in x, y over Finite Field in a of size 5^2
sage: parent(f(k(1),k(1))) 
Finite Field in a of size 5^2