MST Of An Almost Tree
Here is the description of a desired algorithm in Python style pseudocode.

Generate a spanning tree T of G
Put all remaining edges that are not in T in a list L
For each edge e in L:
  Find the cycle C in the graph that is T plus edge e.
  If e is lighter than the heaviest edge e' in C:
    add e to T and remove e' from T


It requires $O(n)$ operations to generate a spanning tree. It requires $O(n)$ operations to find a cycle when you add an edge to a tree and $O(n)$ operations to find the heaviest edge in a cycle. Since there were total $c+1$ edges remaining, where $c+1$ is a constant, the for loop requires $O(n)$ operations. Adding up other operations of small time, we find that the time-complexity of the algorithm is $O(n)$.

The correctness of the above algorithm has been proved in my answer to turn MST of G to MST of G with one new edge. Note that initially T is its own MST.