In a Given we have a numbers in the Range L and R
During each turn, a player can choose any number (regardless of whether or not it was chosen during a previous turn) in the inclusive range L between R and.
The game ends when the running sum of chosen numbers (i.e., sum of all numbers chosen by both players) is greater than K, and the last player to take their turn wins.
So which player win First or Second ?
Depends.
For example, if $L+R > K > R$, then clearly, player two wins, since he chooses $R$, and player $1$ chose at least $L$, so the sum is at least $L+R$.
If $L=R$, then the game is entirely deterministic, and the result depends on whether $\left\lfloor \frac KL\right\rfloor$ is even or odd.
I don't see an easy answer for the general case of $L, R, K$, however.