I know with external flash we can just use better batteries, but what about the built-in flash? Can we reduce its power or give it an external charger? I use a Nikon D5200.
You can't attach an external battery pack or use batteries with higher amperage, but you can set the power lower, either by using flash exposure compensation, if the pop-up flash is in iTTL, or by explicitly setting a lower power value if the pop-up flash is in Manual mode.
Keep in mind, however, that a pop-up flash isn't all that powerful to begin with (guide number: 13m), so depending on your iso and aperture settings, lowering the power to get a faster recycling time may render your flash less than useful for certain subject distances. [Ditto FP/HSS flash]. The flash illumination in an exposure is controlled by ISO, aperture, subject distance and flash power output, so with lower power settings, you're liable to have to use a high ISO setting, a pretty wide aperture, or close subject distances, or any combination of those three things.
The flash recycle time is just a factor you have to get used to with flash photography. Pushing up against it either leaves you with very little power to work with, or running the risk of overheating your flash, and if it's that major a factor to you, then it may be time to consider getting a more powerful light than a pop-up flash or a speedlight, and going for a studio strobe instead.
As far as I can tell from the manual there is no way to directly control the flash power. However if you use a higher ISO then the flash power needed to obtain the same exposure will reduce by the same factor; so for example if you are shooting at ISO 200 then switching to ISO 800 will mean the flash will only use a quarter of the power. Additionally, keeping the aperture as wide as possible will reduce the required flash power (note that shutter speed has no effect on the flash illumination).
In addition to controlling the flash power, you can also use only almost full batteries in the camera. The higher the charge in the battery, the faster the flash will recycle. So going ahead and putting in a fresh 100% charged battery when the one you are using is only discharged to, say, 75% will keep the flash cycling at a higher rate than if you wait until the first battery is at 10%.