Put    $t=1-\sqrt{s}\in[0,1/2)$ so the equation writes 
$$ \Big(1-\frac p2\Big)\, t^p+  \frac p2\, t^{p-1}=2^{-\frac p2}$$

With another explicit change of variables the form $t=\alpha u^\beta$,   for suitable $\alpha$ and $\beta$, and $y=y(p)$, the equation can be put in the form 
$$u+u^q=y$$ 
with $q=q(p)>1$,  that can be solved by series (see e.g. [here][1]).  


[1]: https://mathoverflow.net/questions/249060/series-solution-of-the-trinomial-equation/249098#249098