Short answer: Yes, covid increases your risk of cancer, but it's not a huge risk.

Longer answer: Covid, like HPV, messes with our cells ability to make the p53 gene, which among other things, helps our bodies destroy cancerous cells.

In mild infections low p53 levels lasted 16-24 weeks before returning to normal.

In severe infections low p53 levels lasted past 24 weeks and may not return to normal.

It took a long time to measure the impact of the HPV vaccine, and it will take a long time to meaure the impact of covid and vaccines.

The video compares it to the risk of getting HPV and the Epstien-Barr virus. However I don't think either of those viruses mutate the way covid does, or have the potential to reinfect you with a different variant every several months the way covid does. Also, all three viruses have shown the ability to persist in people's bodies, which increases the risk of cancers and other long term health effects.