add white background

good morning, I want to add a white background to my image which needs to be partially overlapped with other graphics.
I tried with the Live Paint Bucket tool but it turns out to be cumbersome due to the fact that the paths are often open and intertwined...
Sometimes I use the Blob Brush tool but I'm a bit imprecise (overflows).
how do you do it?

pen tool ?

please see image: