If the cooler is working properly there should be no need for this; CPU air coolers radiate heat everywhere to cool the CPU as effectively as possible, which means they require good airflow or that heat would just build up and cooling efficiency would drop.
Liquid coolers meanwhile move the heat to a radiator, which should be mounted as an exhaust to the case, meaning the heat is expelled directly, and shouldn't be able to heat up the case.
Other components on the motherboard shouldn't get so hot as to justify this, and my main concern is that adding this fan could actually be worse for cooling; instead of heat rising from the motherboard and being swept away as part of the exhaust airflow, this fan will be disrupting the airflow of the case by blowing down.
A radiator mounted as exhaust, with appropriate intake fans should be more than enough to get a steady stream of air through the case, which is more than enough to keep everything cool in my experience.
In fact, adding a fan to the pump also increases the height of it, which will clutter the interior more for what I think will be minimal (if any) gains, and eliminates one of my favourite aspects of al-in-one coolers, which is the ability to use them to cool high performance small form factor systems (as they're much more effective than low profile air coolers).