No heat

Car has a OHV engine. Recently replaced water pump due to leak. Before leak, driver reported the temperature gauge in car would get to 50% (normal running temperature) and drop rapidly back to C and cycle again and again. I suspected bad thermostat. Was replaced with a 195 degree thermostat. Car is still getting to fifty percent heat on temperature gauge and dropping, but only dropping to about thirty to forty percent of "normal" fifty percent. The car has had lukewarm heat at best for a while. The supply line to the heater core is hot as is the bypass. The outlet is warm, but not hot. I have not checked the actuator, but when you turn the heat on in the car, it is warm for about ten seconds then gets cold again. Would a flush of the heater core itself solve the problem? I plan on checking the actuator also. Thanks for the help!