  We use Brauer-Manin obstructions to explain failures of the integral Hasse
principle and strong approximation away from infinity for the equation
x^2+y^2+z^k=m with fixed integers k>=3 and m. Under Schinzel's hypothesis (H),
we prove that Brauer-Manin obstructions corresponding to specific Azumaya
algebras explain all failures of strong approximation away from infinity at the
variable z. Finally, we present an algorithm that, again under Schinzel's
hypothesis (H), finds out whether the equation has any integral solutions.
