What current evidence supports the use of hydroxycloroquine used as a prophylactic for (corona)-viral infections?
The only double blind randomised clinical trial I am aware of is this one, examining the use of hydroxychloroquine for the treatment of HIV-1-infected patients.  It finds significant reductions in viral load, but not improvement of CD4+ counts.  How much change could be expected in 8 weeks I do not know.  There are in vitro studies, such as this one looking at dengue virus.

In combination with azithromycin it has been examined for the use in COVID-19 (pre-print, not peer reviewed), but without a control group I find it hard to call it a clinical trial, and there are other criticisms of the work.