The sum is only convergent for $p>1/2$, so for $p=1/2$ you could regularize it by adding a small positive increment: $p=1/2+\epsilon$, $\epsilon>0$.
For $a\gg 1$ you can then approximate the sum by an integral, to arrive at
$$I_{a,p}=\int_0^\infty \frac{1}{(a+x^2)^p}dx=\frac{\sqrt{\pi a } \Gamma (p-1/2)}{2 a^p\Gamma (p)},\;\;\text{for}\;\;p>1/2.$$
This compares quite well with the sum
$$S_{a,p}=\sum_{n=0}^\infty\frac{1}{(a+n^2)^p}$$
as you can see from the plot where both $I_{a,p}$ (blue) and $S_{a,p}$ (orange) are plotted as a function of $a$ for $p=0.51$.
http://ilorentz.org/beenakker/MO/sum_integral.png
This is for $p=1/2+\epsilon$. For the larger values of $p$ you mention no regularization is needed and the sum can be evaluated directly, for example
$$S_{a,3/2}=\frac{1}{32 a^{3}}\left[2\pi ^3 a^{3/2} \coth \left(\pi \sqrt{a}\right) \text{csch}^2\left(\pi \sqrt{a}\right)+3 \pi \sqrt{a} \coth \left(\pi \sqrt{a}\right)+3 \pi ^2 a \,\text{csch}^2\left(\pi \sqrt{a}\right)+8\right]$$