Can I reprocess pickles that I have canned?

I think the question you might be asking is how do I thicken a watermelon pickle after processing it and getting a thin texture.
As I understand it (and I've made preserves many years), it's nit safe to process pickles twice and store at room temperature.
Rather, keep the once-processed pickles sealed at room temp.
Then, as you open each jar, cook the liquid down and/or add ingredients that will help thicken it, then store that one
re-cooked jar in the fridge as you eat it down.
Rinse and repeat.
Then maybe look for another recipe that gives thicker results when you make a new batch next season.

You could, but it would likely reduce the pickles to mush. It might be better to simply cook down the syrup a bit more after you open a jar to eat. I know that would be inconvenient, but it would help retain the bite of your pickles. The only other suggestion, should you decide you really do want to reprocess the entire batch, would be to add a product called Pickle Crisp to each jar, if you can find it this time of year. It's calcium chloride, and helps strengthen the cells of the pickle, or veggie, so it doesn't soften as much. If you can find and use it, it should help retain some crisp in spite of a second round of processing.