Prim's Minimum Spanning Tree implementation $O(mn)$ or $O(m+n \log n)$?
In your step 3b, you have to carry out at most $\deg(u)$ many decrease-key key operation. Thus in total you need $\sum \deg(u) = 2m$ many decrease-key operations. A single decrease key runs in $O(1)$ amortized time (Fibonacci Heaps). Thus the total running time is $O(m+n\log n)$. If you are not using Fibonacci heaps the running time is slower. Something like $O(m \log n )$.

I do not see how you got to $O(m+n\log n)$ by following your arguments. Also its unclear how you can do all the decrease-key key operation for one step in $O(n)$ time.