SWC instant salinity

Can having chlorine tablets in my skimmers affect the instant salinity on my Hayward Swimpure Plus T-cell 15?

It can - and, over time, it can damage your equipment.

Tablets in a skimmer with flowing water create an acidic environment for your skimmer and other pool equipment leading to staining and erosion. And, when the pump is off, the pucks continue to dissolve and, when turned back on, your equipment gets a blast of that acid. Please only use your pucks in a dispenser designed for them.

I was only using them due to my two current T-cells not working.