Those summer annuals will grow and compete until you have a killing frost.  When do you have killing frost?  October? That is a long time for them to compete.

Any chance you have a neighbor who has a sprayer?  Would they be willing to come over and spray this patch for you?

I'd spray it. If you can't get it sprayed, then I would work it to get the weeds killed.  In my opinion, you don't want to give weeds a head start on growing a crop.