Can I keep 1 specific backup on an external hard drive as it fills up and deletes older backups?
Some advice. I always back up my photos separately from Time Machine. You can save your images in a separate folder on the Time Machine drive and Time Machine will not overwrite them (used to do this with Apertures backup vault). 

Also, depending on the space on your Time Machine drive, you could use time machine to backup your external Photos drive as as well.

I would recommend that you have your photo's in at least two locations (which you currently have; however, Time Machine will eventually overwrite old data that is no longer on you hard drive if it needs the space).