Your group $H_n$ is (I believe) called the *wicket group* by Brendle and Hatcher in their paper [Configuration spaces of rings and wickets][1].  They provide a presentation - see Proposition 3.6 of their paper.  They also provide two references that appear closely related; these are Hilden's 1975 paper and Tawn's 2008 paper.


  [1]: https://arxiv.org/abs/0805.4354