If n is even, you can form a partial edge cover with disjoint four-cycles: pick points u and v, and cover all edges coming from u and from v (except for uv) by disjoint cycles. Now recurse, leaving n/2 uncovered edges which are covered by n/4 many more cycles. If n is odd, save edges coming from w for later, and cover the remaining n-1 points and edges, leaving 3(n-1)/2 edges to be covered three at a time by cycles going through w. This gives a minimum of about n/4 + n(n-2)/8 many cycles, with exact numbers to come later.

Gerhard "Bed For Now. Calculations Later." Paseman, 2017.08.29.

If n is even, you can form a partial edge cover with disjoint four-cycles: pick points u and v, and cover all edges coming from u and from v (except for uv) by disjoint cycles. Now recurse, leaving n/2 uncovered edges which are covered by n/4 many more cycles. If n is odd, save edges coming from w for later, and cover the remaining n-1 points and edges, leaving 3(n-1)/2 edges to be covered three at a time by cycles going through w. This gives a minimum of about n/4 + n(n-2)/8 many cycles, with exact numbers to come later.

Edit: Tony Huynh has provided exact numbers, with the construction in the paragraph above an alternate proof for even n. For odd n greater than 3, the cycle decomposition using an extra 3 5 or 6 cycle improves on the method above. End Edit.

Gerhard "Bed For Now. Calculations Later." Paseman, 2017.08.29.