  In 1951, Linnik proved the existence of a constant $K$ such that every
sufficiently large even number is the sum of two primes and at most $K$ powers
of 2. Since then, this style of approximation has been considered for problems
similar to the Goldbach conjecture. One such problem is the representation of a
sufficiently large even number as a sum of four squares of primes and at most
$k$ powers of two. In 2014, Zhao proved this to be true with $k = 46$. In this
paper, we reduce this to $k = 31$.
