Here is my question: do we have any kind of estimate about $p_{k, d}(n)$ the probability that there are at least $k$ prime numbers in a radius of $d$ around $n$?
Do you have any suggestions regarding related work?
For instance, we know that for $n$ there is a prime $p : n\leq p \leq 2n $ (Tchebychev, 1850), meaning:
$p_{1, n/2}(\frac{3n}{2}) = 1, \forall n>1$
Also since it has been shown that there are infinitely many prime gaps at most 246:
$p_{1, 246}(n) \neq 0, \forall n>1$
$^1$ I believe 246 is the smallest, though 2 is a well known conjecture
Which other unsolved problems, have necessary restrictions on the prime gaps? a related question I just got answered. As the comments on your question talk about though, there's not really a restriction. Primorials (products of all primes up to a number) have potentially massive gaps nearby, You can gaurantee all numbers from the primorial plus or minus 2, until the primorial plus or minus the first prime not in the primorial minus or plus 1, are composite for $30=2\cdot3\cdot5$ you get that all numbers in ranges 24-28 and 32-36 are necessarily composite (divisible by a prime in the factorization of 30). Unsolved conjectures, put bounds on d for all k values. Goldbach, has Bertrand's postulate as a necessary condition. Legendre, implies that two primes exists between $y^2$ and $(y+2)^2$ , $y$, a natural number. Grimm's, implies that $d<\pi(n)$ for $k=1$, for almost all ( all but finitely many) $n$. If not then we have a pigeonhole contradiction.