Sounds like you might have a problem.
Does the company install heat pumps? If so, you will then need written permission from them that you can do heat pump servicing on the side. It's not what you do at the company but what the company does anything in line with with heat pumps. A heat pump is just an air con unit - so its going to be hard to argue that is not 'in line with'.
They will be worried about you promoting your own business while you were working for them.
Most business owners and managers are pretty flexible if you talk to them about it. If you have started doing this business on the side and they have "found out" from anyone but you - they probably feel like they can't trust you. Just say you have no idea how things work in NZ and tell them you are happy to give up the extra work. Then give it up / stop advertising. Then in a couple of weeks when you get referred another job talk to them first (before the job starts) and then work through whatever issues they have.
You should be able to come to some arrangement.
If not - you need to decide if this extra work is worth it and quit. Or give it up completely. And when you have a chance to negotiate a new contract then make sure you don't have the non-compete clause.