  Let $E(N)$ denote the number of positive integers $n \le N$, with $n \equiv 4
\pmod{24}$, which cannot be represented as the sum of four squares of primes.
We establish that $E(N)\ll N^{11/32}$, thus improving on an earlier result of
Harman and the first author, where the exponent $7/20$ appears in place of
$11/32$.
