Value of $\sum 1/p^p$
This is OEIS A094289, where they have no information except computations of the value. This suggests the answer "no" to the question "is there any research ..."
What more can there be said about it ?
Essentially nothing. Related series are
Sophomore's constant $\displaystyle C_s=\sum_{n=1}^{\infty}\frac{1}{n^n}$.
Prime zeta values $\displaystyle P(k)=\sum_{p\in\mathbb P}\frac{1}{p^k}$, with $k\in\mathbb N_{\ge 2}$.
No closed form expressions for these constants are known so far.