I digitized some terrific analog slides, but unwisely at times used an incandescent light bulb as my source of illumination and thereby left a very ugly black blotch in the middle of the digitized image.

Conventional photo editing software, like that attached to Picasa, don't work out for me. I don't have the artist's touch to use it.

So, I am looking for software which will allow me to "paint" over an ugly spot as a professional photographer could. Has anyone else had and solved this problem?

Just think of the television commercial for Microsoft Surface Book which shows a user "painting over" an ugly spot.

I see from your profile page that you use a microscope extender to do your digitising.
Since you have presumably successfully digitised other slides using a different light source, is it not possible to re-digitise the problem images avoiding the use of the incandescent light?
If you have an extensive collection of slides it might be worth purchasing a proper slide/negative scanner or a scanner with slide/negative attachments.
You might even be able to outsource the job to a commercial scanning service.
Unless the area to be retouched is an empty monotone region, I doubt that any software can "paint" over it satisfactorily.