I'd prune it hard after flowering so it can grow up the fencing itself. Usually the results are more attractive than trying to weave long vines from nursery into supports. It needs the fencing to climb on though, don't remove it. As it matures it should completely cover it.

Okay, thanks. Does anyone happen to know the name of this Clematis?

Might be Asao. Hard to tell.

Asoa was my thought too. Readably available at the big box stores. If this is the only clem, most likely it is a fairly common but lovely one.

Marie

Kind of hard to tell the type from your pictures but I could go with Asao as well.