The Core Power can be increased by spending Tech Scrap at the MedBay. Every ten level, a new implant slot will be unlock.
Some implants like the Vital Boost scale with the Core Power.
Maximum Core Power can be raised with the implant type Ancillary Core.
Multiplier : Every enemy you execute, adds an extra +10% multiplier to any tech scrap you get from future kills, and this multiplier continues to increase the more enemies you execute until it starts going up by 3% and finally maxes out at 300% . The multiplier resets to 1.0 if you die, use the Medbay or a First Aid Station.
You can store tech scrap when not needed. But the more tech scrap you carry, the higher are the chance to be affected by a Stamina Regeneration buff (randomly attributed after killing an ennemy).
From a personal stand point : I only increase my Core Power when I can improve a very useful implant. Exemple : Aggression Amplifier v.1 to the version 2 ; or unlock a new implant slot.
Getting enough tech scrap to gain a level (filling the tech scrap bar on the HUD) will grant you back one of each currently equiped injectables. Not being overleveled might be a good idea when exploring. If you don't have an injectable equiped or it is already full, you will instead be awarded free energy with a long decay.
Tips : It also works with lost tech scrap. Knowing this, if you carry enough tech scrap to improve your CP and die during a boss fight, you can get back injectables by picking your lost tech scrap.
Also, been near your lost tech scrap regenerates health slowly. The more are on the ground, the faster the regen is.