Sums of four squares of primes

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 $E(N)\ll N^{11\mathrm{/}32}$, thus improving on an earlier result of Harman and the first author, where the exponent $7\mathrm{/}20$ appears in place of $11\mathrm{/}32$.