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).