Maintaining the value of a polynomial over a dynamically updated input
Your idea generalizes as follows: given an algebraic circuit (over the finite field) or Boolean circuit (computing the bit-wise representation of your finite field elements) computing $P$, then maintain the value at each gate in the circuit. When you change the $i$-th bit of $y$, simply propagate that change along the DAG of the circuit, starting from the input $y_i$. If the circuit has size $s$, this takes $O(s)$ time and space. This could be much smaller than the number of monomials (which corresponds to the size of algebraic circuits of depth only 2).