A TRACON Controller working combined sectors due to staffing issues was distracted by false traffic alerts on their display when Tower turned a VFR aircraft into the path of an aircraft on final approach resulting in a NMAC.

Narrative

I was working the north and west position combined during the afternoon rush. My Supervisor was the only other person in the room. He was working south and acting Supervisor. We were working short. Our manager denied overtime and stated that we had to use a trainee for staffing. Aircraft were being switched late consistently and coordination with the Tower was slow. I cleared a LXJ for the visual approach at ZZZ Runway XX. ZZZ was launching several VFR aircraft to the northeast; directly at the arrival. In addition; our radar has been giving false targets and collision alerts for weeks. I climbed an EDV; who was switched late; as the alert was going off. Immediately; I gave a safety alert to the LXJ and canceled the approach clearance. After ensuring that the aircraft was called in; I recleared them. The Tower then turned a pattern aircraft directly into the arrival. The targets overlapped and were separated by about 100 feet. I wanted to leave sick; but there was no staffing to cover it. We are put in these situations daily and this causes extreme stress and fatigue that our management does not address. ZZZ NEEDS a Tower radar display IMMEDIATELY. Our procedures and equipment need to be reviewed.