Relay does not turn ON
One thing that limits what you can do is that the ESP chip is 3.3V and your supply is 5V so you cannot use just a PNP transistor to do what you want. Here is one way to accomplish the task without changing the logic (low = on). 







simulate this circuit &ndash; Schematic created using CircuitLab

Your circuit has two problems - the voltage across the relay coil will only go between about 4.3V and 1.1V- neither fully on nor fully off. That is because it is an emitter follower and has a 0.6~0.7V offset and it cannot amplify the 3.3V coming from the ESP chip. 

The 1.1V is particularly dangerous since it is a grey zone of relay operation and some relays may stay pulled in (depending on temperature, whether they have been tapped, etc.) and some may drop out. The 4.3 means you are not getting full voltage on the coil so it might not pull in at very high temperatures. With the circuit above, the relay voltage will go from about 4.9V to 0V, as it should.