I moved in 2019 for work (>40km). I had moving expenses that year (house sale in Aug. etc). New home bought in June 2020. Can I claim moving expenses in both tax years?

Yes, you can claim moving expenses for years 2019 and 2020 only if you moved into the new home for work or school and the new home's location is 40 kilometers closer to your work/school than the one you were in. If you'd like to know more about claiming moving expenses, here is a link to a great article by TurboTax.