Consider a graph $G$, and inside this graph consider two vertices $i$ and $j$ that are connected to each other, through an intermediate node $v$: $$ ... i - v - j ... $$
We make some local modifications to this graph. Suppose we duplicate $v$, $k$-times and create a cluster of nodes $V$. Each node in this cluster is connected to $i$ with probability $p$ (independent of each other); similarly, each node can be connected to $j$ with probability $q$. With this probability of having a path between $i$ and $j$ is $1 - (1-pq)^k$.
Suppose we randomly with prob. $r$ add edges between any node-pairs in $V$. What would be the probability of having a connection between $i$ and $j$?
Apparently the problem is called 2-terminal reliability problem (thanks to @Salomo for pointing it out). To the extent that I searched there is no analytical closed form.
An easier solution is to use the phase transition results. For example, if $r > \frac{\log n}{n}$ $V$ will almost surely be connected, which means that the probability of having a path is $1-(1-p)^k(1-q)^k$ and (almost surely) independent of the exact value of $r$.