  Let k>2 be a fixed integer exponent and let \theta > 9/10. We show that a
positive integer N can be represented as a non-trivial sum or difference of 3
k-th powers, using integers of size at most B, in O(B^{\theta}N^{1/10}) ways,
providing that N << B^{3/13}. The significance of this is that we may take
\theta strictly less than 1. We also prove the estimate O(B^{10/k}), (subject
to N << B) which is better for large k.
  The results extend to representations by an arbitrary fixed nonsingular
ternary from. However ``non-trivial'' must then be suitably defined.
Consideration of the singular form x^{k-1}y-z^k allows us to establish an
asymptotic formula for (k-1)-free values of p^k+c, when p runs over primes,
answering a problem raised by Hooley.
