There are two ways to introduce front grip in a kart if it rains. The first is a very easy fix; put on wets. The second takes a little practice to get the set up right, and a lot of work with the front end of the kart. First off, a quick fix would be to widen up the front. Long hubs, more rings, anything it takes to get that front as wide as possible. The reason the front pushed is probably because the rear had more grip in the situation (due to your sitting position, axle softness, narrower tires so less surface contact etc….) and therefore the front did not have enough, leading to understeer. Note that if you induce more grip in the front, you may get to much front grip and have problems with over-rotating it into a corner. Therefore you always have to counter what you do to the front with rear set up changes as well.
Other than set up, one thing I wish I learned earlier was to use your body in the kart. If you lean backwards and to the outside of the corner, the loaded wheel (in a left turn its the right rear, a right turn its the left rear) gains grip and improves side bite, which will then increase rear grip. It will also give you more stability mid corner and exit.
And on the topic of kerbs and rubber, stay off ALL kerbs on the ENTRY to a corner. If you break on a kerb, you will lose the kart. There are many different ideas on what to do with the rubber, and how to change your line in the wet. A basic idea to have is to “cross the rubber”. I find that breaking slightly to the inside of where you usually break, and then getting the kart to the outside of the corner works best. Basically, reduce all contact with the rubber on the track to a minimum. In other words, instead of outside inside outside, its inside outside middle. On entry, don’t go to far inside, but more inside than your usual breaking point.
This is because as was stated, when rubber gets wet, it does not give much grip. This is for several reasons. One, being that the surface is smooth, so if you break on it while wet, the water gets jammed between the wheel and the surface and you aquaplane. Also, when it rains, the rubber can release oils that the tire naturally holds. When this oil comes out, it gets very slick. This isn’t such a big deal in the full wet, but it can be problematic in drying conditions.
Hope this helped a little, but rain racing takes a lot of practice, and usually some luck too.