Ok first, you need a battery for all this to work. The solar charge controller won't drive into nothin', that is not what they are designed to do.
Its full name is "solar battery charge controller". Size the battery any way you like, just make sure the charge controller knows what kind it is, so it sets the proper charge curves and doesn't overcharge it/boil it dry/make it catch fire (lithium).
Once you have a battery in the system, paralleling in another battery charger is not a problem, be it wind, AC mains, or another solar CC.
Now, you will need to manage the case where the Arduino calls for the pump to turn on, but the solar panel is not making enough power to successfully power the pump. This will not magically sort itself out.
If you did not have a battery, it'll "crow bar". The pump would effectively pull the solar voltage to near zero, which would cause the Arduino to shut down, turning off the pump and causing solar voltage to recover. The Arduino would reboot, try again, and fail again. Repeating. So a crow-bar condition.
If you do have a battery, then the Arduino will successfully spin up the pump, but, the solar won't be enough and this will be depleting the battery. If your usage runs the battery flat, see crow-bar above.
If you have a battery and the Arduino monitors battery voltage, it can defer operation until the Arduino knows the battery is sufficiently charged for the work needed. At this point you don't need to care if the solar is up, the battery will carry the "buffer". Make sure to shut the load off before the battery is depleted enough to create a crow-bar condition.
Note that if the battery is lead-acid, it should be managed to stay within 60-100% state of charge. Depleting a lead-acid below 60% of max will rapidly age the battery. It probably will be, since they are a happy intersection of "cheap" and "well-supported by every charge controller ever made".
Power to the Arduino itself should be taken as close as possible to the energy source (e.g. battery) — even if that means a long, redundant cable, especially in that case actually. Do not share a supply cable with motors or pumps or other heavy loads!!! I have seen many installations where the startup current of the motor causes enough voltage drop down the shared supply cable to cause the Arduino to lose power and reboot. That is quickly cured by running a redundant V+ and V- power cable just for the Arduino.