Whatever it is, it looks very cramped in that pot - perhaps it needs re potting in a much larger one.

Looks a bit like a honeysuckle of some kind. If it's possible I would cut it back, remove it from the pot and give it a new home in the garden where it might be happier.

Agreed, Honeysuckles are very prone to mildew when they are dry at the roots and in a pot they often get very dry at the roots, even if watered regularly.