This is a complex issue. It is not just a matter of someone staying overnight. But you refer to a partner, which generally implies some sort of semi-permanent arrangement. If you are claiming as a single parent and have a partner then you must end the claim & make a new one as a couple.
Go here:
http://www.hmrc.gov.uk/manuals/ccmmanual/ccm15 040.htm
Read it carefully & follow the links the page gives. You will then be able to assess the various factors against your situation.