How to determine whether a graph has l cycles of length k exactly?

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Given integers$ l\ge 0$, $k\ge 3$, and a graph, I want to know if it contains $l$ $k$-cycles exactly. I could enumerate all cycles of length k using the link below and count them, but this might not be very efficient. - subgraph_search_iterator-gives-duplicate-subgraphs

G=graphs.OctahedralGraph()
H = graphs.CycleGraph(3)
print( sum(1 for _ in G.subgraph_search_iterator(H, induced=False,return_graphs=False) ) )
L = { tuple(g.edges(labels=False,sort_vertices=True,sort=True)) for g in G.subgraph_search_iterator(H, induced=False) }
print(len(L))

In an import case for me, determinie if a graph has a unique $k$-cycle (when $k=1$), I don't need to list all cycles and then check; instead, obtaining two cycles would suffice, and the algorithm stops.

Similarly, such questions can be asked about subgraphs