Yes, once you get some CYA in there, the chlorine will last longer. How much CYA did you add a few days back? I wouldn't let the cl drop to zilch or even 0.5 or you may end up with algae. Until you get to the point where you have a CYA reading of around 40ppm, you should be testing and adding bleach at least twice per day and maybe even in the afternoon if you are home, and each time add enough bleach to get back to 5ppm. Each quart of 6% bleach will add 2ppm.

BTW -- until you get a better kit, you can force your kit to read higher by using a dilution method described below. It is not super accurate, and not meant to replace buying a good kit, but it is better than nothing.

Testing Without a Good Kit

4 hours a day of pump time is not enough. Typically, most pools need 8-10 hours per day. You can split into two 4 or 5 hour sessions if you want. When you add the CYA to the filter, though, you'll need to run it 24/7 for at least 4 or 5 days while it is dissolving.