Is there any current research on Gödel machines? It seems that the last article by Jürgen Schmidhuber on this topic was published in 2012: http://people.idsia.ch/~juergen/goedelmachine.html
This paper Can Machines Design? An Artificial General Intelligence Approach (2018, presented at AGI-18 and published in the related proceedings here), which proposes the design Gödel machine, may be useful to you.
After a quick search, I have not found other relevant papers, so I suppose that the research on GMs is currently not very active.