A Waring–Goldbach type problem for mixed powers

Let PrPr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, it is proved that for each integer k   with 4≤k≤54≤k≤5, and for every sufficiently large even integer N   satisfying the congruence condition N≢2(mod3) for k=4k=4, the equationN=x2+p12+p23+p34+p44+p5k is solvable with x   being an almost-prime PrPr and the other variables primes, where r=6r=6 for k=4k=4, and r=9r=9 for k=5k=5. This result constitutes an improvement upon that of R.C. Vaughan.