I imagine that Manny Hawthorne will be happy to have your participation in the one-name study that he started, Derek. It looks like he hasn't gotten very far off the ground with it yet -- and from his activity log I infer that he may be on holiday. We collaborate at WikiTree, so this single name study should include everyone who's interested in focused study of this family name.
Because there are a number of variant spellings of the name, your intention of including Hawthorn in this study seems very appropriate. It seems to me that other spellings, such as Hathorn and Hathorne, also ought to be included. The Hawthorne name study page can accommodate three tags, only one of which is currently in use, so it might be a good idea for you to be bold and insert a couple of variants. (If he's bothered by that, your changes can always be reversed.)
It's likely that the name has multiple origins, so it may be desirable to document different lineages of the name. If you have a good story to tell about your Hawthorne line, you might want to create a free-space page to tell it -- and add your page to the Hawthorne Name Study category. You could also categorize the profiles of your Hawthorne family members.