  Let $f(z) = \sum A(n) n^{(k-1)/2} e(nz)$ be a cusp form of weight $k \geq 3$
on $\Gamma_0(N)$ with character $\chi$. By studying a certain shifted
convolution sum, we prove that $\sum_{n \leq X} A(n^2+h) = c_{f,h} X +
O_{f,h,\epsilon}(X^{\frac{3}{4}+\epsilon})$ for $\epsilon>0$, which improves a
result of Blomer from 2008 with error $X^{\frac{6}{7}+\epsilon}$.
  This includes an appendix due to Raphael S. Steiner, proving stronger bounds
for certain spectral averages.
