No you are not correct, as David has pointed out, UC is different from ESA and therapeutic earnings. The rules to claiming LCWRA are no different than claiming ESA support, the form is the same apart from the name, ESA50 now called UC50.
We would advise you to seek advice from your local Welfare Rights Organisation; advicelocal.uk
Gary
Nothing on this board constitutes legal advice - always consult a professional about specific problems