LM2940CT-12 output measuring 9.6V when relay used
Three main issue. 


Battery voltage under load. Any battery will see a drop in voltage when a load is applied. It's worse on smaller cells with high esr, but it happens even in bigger batteries. The true voltage of the battery can only be known when you measure it under load. 
The regulator has a minimum input voltage to output voltage difference. This is called a drop out voltage. If Vin - Vout is less than the drop out, it will not regulate properly. This needs to be under load! The LM2940 will have a drop out of 0.5V to 1V at its max current


1 &amp; 2 is why the fan is dropping the voltage down. 


The max current of the regulator is being exceeded. The LM2940 has a max of 1 Amp. This will cause a higher drop out, heat issues and regulation problems. 


1, 2, and 3 is why the pump is bringing the voltage down to 9 volts. 

Also, not sure if you are heat sinking the regulator, and a input capacitor is always helpful.