dont worry about that. just set up a new material for the handle. i presume you just want a plain ceramic cup texture for that part of the cup? any way press ctrl i to invert the current selection. (the selection in your image. you maybe changed it by now, but thats the image i’m working with.)
ok invert selection and choose your new material from the material slot. in edit mode press assign which is a button just under the texture preview panel. now you have you image mapped to your uvs and a diffuse texture applied to the handle area. that should be all.
oh just FYI you’ll often get texture bleed across your faces when you have a subsurf modifier on the model. you can fix it with control loops or just dont worry if it is not making the model look wrong