Assuming the Waring-Goldbach problem (see https://en.m.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem) has a positive solution, can we reduce the number of terms $t$ to some value $t'$ by allowing the $t'$ exponents to take $1<u<t$ different values?
Indeed it seems to me that requiring the exponents to be all the same is too strong a constraint preventing $t$ to be small. Can we for example hope to get that every large enough integer $N$ fulfilling some congruence condition modulo $t'$ is the sum of $t':=t+1-u$ terms of the form $p_{i}^{k_{i}}$ with $\displaystyle{\sum_{i=1}^{t'}k_{i}=tk}$?
