I could find one study:
https://www.medrxiv.org/content/10.1101/2020.05.02.20080036v...
Conclusion: This study provides the first in vivo evidence that zinc sulfate in combination with hydroxychloroquine may play a role in therapeutic management for COVID-19.
This one is underway: https://clinicaltrials.gov/ct2/show/record/NCT04377646