That's normal. It's an issue with the Ps4 kicking out 3rd party controllers, not an issue with the Hitbox. If you have the most recent drivers (or at least, drivers from within the last 2 years or so), there is a shortcut to manually "reset" the Hitbox. I think it's RB+Select, but I'd have to check. That causes it to disconnect for a second or two, and then reconnect. I just do it between matches (even on PC, since I just do it without thinking, now). I'm responding from a shitty laptop on shitty wifi, but I can try to find links to the drivers and shortcut info for you. It was a hassle initially, but once you get set up and get in the habit of resetting, it's no problem.
You can also just unplug and replug the cord from the back of the hitbox.
[Edit]: Here is info on how to update the firmware (which should give you access to the shortcut):
https://www.hitboxarcade.com/blogs/ps4-firmware-update. Still looking for info on the actual shortcut.
[Edit] Ha, just found my answer in another Hitbox thread. Here you go:
You need to make sure your firmware is totally up to date. PM me if you need help with that. Once it's up to date, the combo is R2+Select, then after a second or two you need to press the Home button and select your profile. It takes about 3 seconds in total.