So what's the status of this technique/therapy in psychology? E.g are there meta-analyses showing that it's effective? And how is the quality of primary research assessed in this area? E.g. is the sub-field reasonably free of publication bias (which nowadays is also assessed in the better-off meta-analyses)?
First off, I am not familiar with this therapy. There seems to be 251 hits on Scholar (excluding citations and patents) at the time of this writing with a search for "Havening Technique'. Of these 251, there seems to be one single journal article from Gursimran et al. (2015). The publisher of this 'journal', i.e. the Health Science Journal (Insight Medical Publishing) is listed on Baell's list and hence a suspected predatory publisher.
In all, this Havening technique can safely be considered as a technique that has not been described in the scientific literature.
I am deliberately leaving out the reference to the paper, as I regard it a source, not a reference