Please help improve my water pump

You need both more battery and to recharge it, more panel.

Our question is, How many cloudy days to you need this to run for ? Generally systems are designed for 2-4 days of clouds, and day 3, the generator is started so as to not overly deplete the battery. If you say 8 days, you will need a much much larger system.

8 watt pump That would not need a very large inverter, a 50W inverter is about the smallest you can get. For things using a motor, you should use s pure sine wave inverter, it allows the pump to run cooler. High wattage inverters (over 300w) will self-consume more power internally, than your pump does. Replacing your inverter with a high efficiency pure sine wave, may be all you need to do.

8 watts x 24 hours = 192 watt hours
50ah battery @ 12V = 600 watt hours stored, 50% are usable. If you have already drained the battery to shutdown, several times, the battery has had it's lifetime shortened quite a bit, so start the budget for a replacement battery

To get a solid 3 days, you need 1200 watt hours of battery, if you don't want to damage the battery on day 3.

How can you prevent a battery from draining to shutdown? Some controllers have a function to prevent this?

Most charge controllers will turn off the battery at a voltage below 11V which is not good. There are other external Low Battery Relays that can be programed higher which will help extend the batteries life.