Yesterday I came up with an asymptotic expansion for the partial sums of the prime zeta function $$\mathcal P(x)=\sum_{p\le x}\frac1{p^s},\quad p\in\Bbb P$$ with the extra constraint that $s\in\Bbb Z^+\setminus\{1\}$. This was done by first considering the sum $$A(x)=\sum_{n\le x}\frac{a(n)}n\log n=\log x+\mathcal O(1)$$ where $a(n)=1$ if and only if $n\in\Bbb P$, and using Abel's summation formula to give $$B(x)=\sum_{n\le x}\frac{a(n)}{n^s}\log n=x^{1-s}A(x)+(s-1)\int_1^x\frac{A(t)}{t^s}\,dt=\mathcal O(1-x^{1-s}).$$ Abel's summation formula was used once more to give $$\mathcal P(x)=\frac{B(x)}{\log x}+\int_2^x\frac{B(t)}{t\log^2t}\,dt=\mathcal O\left(\int_2^x\frac{1-t^{1-s}}{t\log^2t}\,dt\right)$$ as an asymptotic expansion (this can also be written in terms of the exponential integral function). However, this is not too meaningful, as the RHS consists of $\mathcal O$ terms only.

Are there better asymptotic expansions of $\mathcal P(x)$ (in literature or otherwise) that include non-$\cal O$ terms as well?