Green algae? Mustard algae?

[AE340B]

I have a Hayward DE filter with a Hayward Super Pump. I also have a Rheem electric heat pump for heating the pool. Given all this equipment, can I pour bleach right into the skimmers, or do I need to remove the solar cover and distribute bleach into the water every time I need to add it? I'm worried that having the bleach circulate through the filter, pump and heater in such high concentrations will damage this equipment somehow. The heater has a copper coil, not titanium, unfortunately. Perhaps even more importantly, the bleach will be going through my Hayward Chlorinator with 3" pucks in it. Will this cause some sort of chemical mismatch? I know there are some types of chlorine you can't put into the chlorinator.

As for the solar cover, it's a necessity as I pay the highest electric rates in the country (0.19 KwH !!) , and it saves me over $1000.00 per season in electric to use it. It is, however, a major hassle to use, as I have an unusually large "L" shaped pool and the cover has to come off and on manually in three separate pieces. Any info on the perils, if any, of adding the bleach right through the skimmer would be appreciated. Thanks!