Consider the following sum of function of primes:

$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$

Here $p$ runs through all primes and $e$ is Euler's constant.

We can see that the sum diverges.

I have following questions :

Is possible to regularize this sum ? If yes, how to do so?

Any advice about going around this is welcome. Any insights in such type of problems is/are also welcome.

Related: A question on assigning finite values to divergent sums involving expression of primes

On modified Euler product

I used the explicit formula for prime counting function $\pi(x)$ and integrated the given prime function with measure as $\pi(x)$. But i couldn't deduce an exact value. (If it is of any help, the answer I'm expecting is $(\gamma-3)$)