HamCycle to HamPath reduction
First of all I think that the direction of the reduction of your question is from Hampath to Hamcycle (you prove the NP-hardness of Hamcycle reducing the NP-complete problem Hampath to it).

Now given an instance of Hampath: $G,s,t$: 

1) there is an edge between $s$ and $t$
2) there is not an edge between $s$ and $t$

In both cases the extra node $u$ is needed in order to force $s,u,t$ to be consecutive nodes in  the Hamiltonian cycle. 

The new graph $G'$ with a new node $u$ and new edges $(s,u), (u,t)$ (in the first case you split the original edge) will have an Hamiltonian cycle iif the original graph $G$ has an Hamiltonian path from $s$ to $t$.

You can "play" with these two graphs: