Why is my 220v - AC/DC 3.3v transformer cutting power when relay comes on?
The relay will draw about 132 mA steady state. V = IR, so I = V/R = 3.3/25 = 132 mA. The ESP8266 can draw up to 170 mA, on an intermittent basis. The radio goes in and out of sleep mode, thus you may have intermittent issues if your 3.3V supply is not able to provide sufficient current.

It looks like the supply you chose is rated for 700mA, so that should not be a problem. The issue may be related to the inductance of the wiring. I see that you have a capacitor, C2,  in the schematic for that purpose, but you need to be sure it is very close the the module leads of the ESP module or it will not help. Also the inrush current of the relay can be much higher that the holding current, and it will draw more current when first turned on. Again a capacitor very close to the emitter of Q1 to ground will help. 

Bottom line, your problem may very well be layout related, not schematic related.