It was shown by \textit{K. Kawada} and \textit{T. D. Wooley} [Proc. Lond. Math. Soc. (3) 83, No. 1, 1--50 (2001; Zbl 1016.11046)] that every sufficiently large integer \(n\) is representable as \(\sum_ {1 \leq i \leq 14}p_ i^ 4\) in primes \(p_ i\), provided it satisfies the congruence condition necessary mod 240 if the primes are not 2, 3 or 5. For the corresponding case of Waring's problem, \textit{R. C. Vaughan}, improving on work of \textit{K. Thanigasalam} [Bull.\ Calcutta Math.\ Soc. 81, No. 4, 279--294 (1989; Zbl 0641.10037)], established a corresponding result [Acta Math. 162, No. 1/2, 1--71 (1989; Zbl 0665.10033)] for sums of 12 fourth powers of integers, and \textit{K. Kawada} and \textit{T. D. Wooley} [J. Reine Angew. Math. 512, 173--223 (1999; Zbl 1005.11046)] reduced the number 12 to 11, except that their treatment did not cover the case \(n \equiv 11 \bmod 16\). The authors allow just one of the fourth powers to be a power of an integer as opposed to a prime, and are then able to work with one fewer power than Kawada and Wooley did for the corresponding Waring-Goldbach problem. The authors' enunciation is that any sufficiently large number \(n\) is representable as \(m^ 4+\sum_ {1 \leq i \leq 12}p_ i^ 4\), subject to the corresponding condition mod 240, but this is done by establishing an asymptotic formula for the number of representations. Besides drawing on references to standard texts by K. Prachar, E. C. Titchmarsh and R. C. Vaughan, and to results from the papers of Thanigasalam and of Kawada and Wooley, the authors use \textit{D. R. Heath-Brown}'s generalisation [Can. J. Math. 34, 1365--1377 (1982; Zbl 0478.10024)] of Vaughan's identity to deal with certain exponential sums over primes.