In most cases, whenever a printer won’t pickup paper, it is usually because of the pickup rollers. I would recommend going to hp.com and searching for your specific printer model. Look for the part lists and find the part number for the pickup roller. Note, most printers have at least two trays. One is where you load most of your paper and the other for the manual paper tray. You want to make sure you change the rollers for the tray that is giving you the most problem.
You can also verify this to be a problem if you touch the rollers. If they’re really smooth, it won’t grab the paper. Rollers should be almost tacky.
Finally, you can try different paper or fanning the paper so that it’s not sticking to one another.
Good luck.