Replacing either the light socket or the whole recessed light fixture
This looks like a box for incandescent/CFL lighting that is not rated for insulation contact (IC). 

It might not be easy/cheap to replace it. Even if it's a socket you might find it hard to find a replacement part. You will need to pay with either time (DIY) or money to fix it. I doubt contractors will be interested in repair of such a thing, too time-consuming for them and you are unwilling to pay "too much money". Where I live an electrician visits starts from $200

For the same money consider hiring a handyman to do remove the box, add some framing to support a new drywall patch, patch the seam, match-paint, paint the ceiling, cut a round hole and install an IC rated can like this.
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That would probably also cost you about $200