  We prove that for operators satistying weighted inequalities with $A_p$
weights the boundedness on a certain class of Morrey spaces holds with weights
of the form $|x|^\alpha w(x)$ for $w\in A_p$. In the case of power weights the
shift with respect to the range of Muckenhoupt weights was observed by N.~Samko
for the Hilbert transform, by H.~Tanaka for the Hardy-Littlewood maximal
operator, and by S.~Nakamura and Y.~Sawano for Calder\'on-Zygmund operators and
others. We extend the class of weights and establish the results in a very
general setting, with applications to many operators. For weak type Morrey
spaces, we obtain new estimates even for the Hardy-Littlewood maximal operator.
Moreover, we prove the necessity of certain $A_q$ condition.
