On Vacation and need help! Hvac help cooling issues

I was kindly allowed to stay at a family friends vacation home in Florida this week. We arrived Saturday to find the house at 87 degrees. I found the outside unit was not operating. After some inspection i found that the 240v relay was not engaging. When i manually pushed the plunger it worked. However when the thermostat temp is reached i have to disconnect the prime power fuze wait for relay to click then return 240 v power. Whats the issue here and am i running the unit in a way that i could cause damage?