It depends on the type of ESA. If you are on contribution based ESA based on National Insurance contributions it doesn’t matter if your wife works or not because it is not means tested. Being in the support group means that if you start on contribution based ESA you stay on it beyond the 12 month limit imposed on people in the WRAG. If however you are on income based ESA then all the money coming into the household is considered, partners earnings, savings etc. if your income is too high then you would not be paid any money although you would still be technically eligible for ESA.
How to know if you are on contribution or income based? It should have said on your original award letter but who keeps them? What were your circumstances prior to getting ESA? did you go straight onto it from a long period of full time work? If so you are likely on contribution based. But the fact that you mention telling them that your wife works gives me pause, they would have no reason to ask about your partners circumstances if they were basing your eligibility on NI.
As Lenney suggests CAB will be able to give you clarification.