The white bumps almost sound like Ich. Due to an uncycled tank undoubtedly.
Have you heard of the nitrogen cycle before? It's where bacteria build up the filter and on surfaces in the tank. When the tank is first set up, this bacteria is not present yet. These bacteria help filter out all the bad stuff in your tank like ammonia. Ammonia comes from fish poop and dying plant material if you have live plants. The bacteria then convert it to NitrIte, a slightly less toxic form but still very toxic to fish. From there, the bacteria will convert it to nirtAte. NitrAte is much loss toxic to fish and only becomes toxic if let go into toxic levels (80+ ppm
). This process is known as the Nitrogen Cycle. Many people new to the hobby do not know of this process. It is simple and easy. A few weeks and the cycle is done and your fish can live healthy and happy. If you go to the Getting Started section of the forum and click the sticky at the top it will tell you all about cycling your tank properly.
Next thing, I think I read that your tank is half a gallon? If not please correct me. That is way too small. This is a common misconception generated by big box companies. Bettas need at least 2.5 gallons preferably 5 gallons. They also need a heater as they prefer temperatures around 78-80 degrees F.
Next, you Betta likely has what is known as "Ich". Ich is not present in nature because of the abundance of water fish live in. In the home aquarium they have the water contained in the 4 or more walls around them. Ich is a free floating parasite that is often caused by stress and weakening of the immune system in a fish. Your tank being uncycled likely caused this stress. If you don't have a heater, get one. Crank the heat up to 86 degrees F. This will greatly increase the life cycle of Ich and cause it to pass much quicker. You should also do water changes every day/ every other day.
I'm going to assume you don't have a liquid test kit. These are vitally important for knowing what's going on inside your tank. It is crucial to have one during the cycling process. I would invest in a lot of you do not have one. An API master test kit runs about $30 retail and sometimes cheaper online. It's great and accurate and lasts a long time. Test strips are very inaccurate don't waste money on them.
Hope this helps!