The easy part first! When you change address, you must make a new application for Rent Supplement with the CWO who deals with your new address. You will need new application forms even if you're still dealing with the same CWO.
Re your question on continued eligibility:
It's a bit complicated, but as a very rough guide -
The first 75 euro of assessable part-time earnings is disregarded. If you are earning less than this at the moment, any increase in assessable earnings up to 75 euro will not change your Rent Supplement entitlement.
If you are currently earning over 75 euro per week, then any increase in wages earned will reduce your rent supplement by three quarters of the increase (assuming there is no change in your JA payment). If your JA reduces, the reduction in rent supplement may be less.
For example, if your assessable earnings are currently 75 euro pw, every 10 euro extra assessed as earnings will reduce your rent supplement by 7.50 (assuming your JA remains the same).
It would be best to ask your CWO who will have the exact figures relating to your situation and can best advise you.
You might find this document useful.
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