For integral weight $k$, I believe that the best result is due to Rankin (1990):
$$ \sum_{n\le X}a(n)\ll_\epsilon x^{1/3}(\log x)^{-\delta+\epsilon},
\qquad \delta:=\frac{8-3\sqrt{6}}{10}\approx 0.065.$$
On the other hand, Jutila (1987) proved that in square mean the sum is of size $\asymp x^{1/4}$, so the exponent $1/3$ cannot be lowered to $1/4$.

For half-integral weight $k$, I don't know the best result from the top of my head, so I skip that part for the time being.