Reduction of Hamiltonian cycle to Hamiltonian cycle & clique
For NP completeness, two things must be demonstrated.  


That a solution to the problem can be verified in polynomial time (usually easy). 
That the input to another NP Complete problem can be formatted as an input to the potentially NP complete problem (also that a solution from the potential NP Complete problem can be turned into a solution to the NP complete problem) all in polynomial time.


It looks like you are trying to go the wrong direction with your question. You are asking about inserting a Clique into a graph, but you only need to concern yourself with a graph that would be sent to a HP solver and modify the inputs so they are appropriate for a HP+Clique solver.  The HP part of the candidate problem doesn't require any modification - it just takes an undirected graph G(V,E).  A clique takes an undirected graph G(V,E) and a goal, k.  So you need to define a k that is in $G_{HP}$.  what if you set k=1?.  This way, the only graphs that your solver will fail to find a solution for are those that don't have a HP since every node is a clique with itself. This takes O(1) to define k=1. 

Now, you just need to modify the output of the HP+Clique so it is the same as the output for the HP problem.  e.g. If the candidate problem produces a solution, you would only return the path part and drop the clique part.   If a solution is not found, you return NO.