The presence of pus certainly shows that there is an infection. You didn't state if the pus comes out from the nail bed, but if it is, its probably a cuticle infection you have got.
You need to use hydrogen peroxide, which decimates the germs. You could use alcohol, even though it burns.
You could make a dressing, so that there is less exposure to water, and germs.
Make sure you change the dressings at least twice a day, and allow some air to blow on it for some time.
Then, you could use cuticle creams subsequently to massage your cuticle.
I have recently had an infected finger. I was told it was a paronychia.
On the photo, to me, it looks more like a blood blister. If it were me I would ***** it with something sterile and release that fluid build up. I doubt this is what any medical person would advise though.
My GP gave me magnesium sulphate cream to draw out the pus. He said to put the cream on and to cover it with a plaster.
That helped a little but it still hadn't improved much so he gave me some crystacide (hydrogen peroxide) cream to put on it. That didn't seem to achieve much either.
Somewhere in all that I was put on a couple of courses of oral antibiotics.
Eventually he had me take a culture of the pus, etc and it came back with both bacteria and fungus. He then gave me an anti-fungal cream. The swelling and pain have gone down but it still isn't 100%. I also had an ultrasound done to make sure there wasn't anything in it.
Mine doesn't look anything like yours does though so maybe it is something totally different.