Structural parametrization for weighted vertex cover
Find the set $L$ containing the $\ell$ extra edges (actually, find any set $L$ of $\ell$ edges such that $G-L$ is a tree).
Let $C$ be an optimal solution.
For each edge $(u,v)$ in $L$, guess whether $u \in C$. If you guess &quot;yes&quot;, add $u$ to a set $X$, otherwise add $v$ to $X$.
(Notice that it is possible for both $u$ and $v$ to be in $C$, but we do not need to explicitly consider this case).
Overall, there are at most $2^\ell$ distinct choices for the $\ell$ guesses.
At least one set $X^*$ among the guessed sets $X$ is a subset of $C$.
Given $X^*$ you can find the minimum weighted vertex cover in poylnomial-time by considering $F=G-X^*$. Since all the edges in $L$ have an endpoint in $X^*$, $F$ is a forest. Then you can find a minimum weighted vertex cover $C_i$ for each tree $T_i$ in $F$. The set $\cup_i C_i$ is a minimum  weighted vertex cover for $F$, and $X^* \cup (\cup_i C_i)$ is the sought solution.
You could also directly find a minimum  weighted vertex cover for $G-L$ with the constraints that the vertices in $X^*$ are forced to be in the cover. This can be done, e.g., with a straightforward modification of the dynamic programming algorithm for trees.
Another option is setting the weights of the vertices in $X^*$ to $0$ (or to $\sum_{v \in V}$) making them essentially &quot;free&quot; to select.
Since you don't actually know $X^*$ you can run the above procedure on each guess $X$ and then select the cover of minimum  weight.