Waring-Goldbach problem: Two squares and some higher powers

Let Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that for every large odd integer N, and 5≤a≤8, a≤b, 1360<1a+1b≤13, the equationN=x2+p2+p13+p24+p35+p4a+p5b is solvable with x being an almost-prime Pr(a,b) and the other variables primes, where r(a,b) is defined in the Theorem, in particular, r(6,7)=5. This result constitutes an refinement upon that of J. Brűdern.