HP laptop touchpad jumps around and clicks on its own

does it only go mad when the charger is plugged in? some HP laptops get a jumpy cursor from a cheap or poorly earthed charger - try it on battery for a while. if it calms down, it's the charger, not the touchpad. also worth turning on the option to switch the touchpad off when a mouse is connected, if you use one.

Check the touchpad is sitting flat before anything else. If it looks raised or the click feels stiff, the battery underneath may be swelling and pushing it up, which gives exactly this - phantom clicks and a cursor that wanders. That one needs the battery replaced soon, don't ignore it. If it's flat, it's usually palm contact. Settings > Bluetooth & devices > Touchpad, drop the sensitivity a notch and turn off "tap twice and drag". Updating the touchpad driver from HP's support assistant fixed it on one I had.