The Klingberg contract is complicated because it was signed mid year and includes a signing bonus. This is my understanding:
1. Klingbergs annual pro-rated salary is 1M. So if he signed it before the season he would have been owed the full 1M. However because he signed it with 91 days left it would be 91/192 or 47.4%. That's 474k in cash that he would be paid which is basically paid out as he plays the game, not as a lump sum.
2. Klingberg had a 350k signing bonus. This amount is paid in full. So if the Oilers are under the cap, the full 350k is added to the 474k. In total that would be 824k in cash paid out to the player.
3. This is different than his annual cap. His annual cap hit is the 1M annual salary plus the signing bonus of 350k. However, since the signing bonus was done for half the games, you essentially have to double (now 700k) if it was an "annual amount". That makes the annual cap hit 1.7 (1M salary plus 2x the signing bonus of 350k).
4. LTIR is a whole other ball of wax. LTIR doesn't increase a teams cap, rather it's supposed to "make teams whole". So when it comes to these mid-season contracts it actually penalizes you if you try using LTIR. So instead of the 824k, it uses that full 1.7M annual amount.
Hope that helps!