Possible Relay Overload
The switching of the pump causes much EMI (ElectroMagnetic Interference). This can be conducted, like Wouter suggests, and then an easy way to test is indeed temporarily use a battery, so that the interference can't come in through the power supply. If it works OK on a battery you'll have to decouple the power supply better. Filters! You can use a mains EMI filter, like this one, built into the socket:  



Also decouple input and output of the power supply with capacitors, and the power connections of the microcontroller and other ICs. Use 10 &micro;F parallel with 100 nF, as close as possible to the power supply pins.

If the battery doesn't help then the EMI is radiated. Also here decoupling with capacitors should be the solution.