How is it possible that transactions can be added to a block currently being mined?
You're thinking that mining a block is a single long computing process, like computing the trillionth digit of pi or something like that. This is wrong. Each attempt at solving a block takes almost no time at all, and each attempt is independent of the others. Miners make many attempts every second. Each attempt has a very small chance of solving the block. It's a lottery, not a race. Modifying the block between attempts isn't a problem.

Another common misconception related to this is that everyone is working on the same block. This is also false. Every miner works on a different block.