My horses core flashes red

[jacks1son]

 What is the What is the significance of the flashing red cores?

[Moosh89]

You need to feed, give medicine or stamina tonics. While on or near your horse, hold L1/LB, press R1/RB to tab over to "Horse" and move the right stick to the Left (9:00) (and while holding L1/LB and the Right stick to the Left you can tab through available feeds using L2/R2/LT/RT) and release L1/LB to feed your horse your selection. I like to use Hay or Wild Carrots as they give large core boosts to both Stamina and Health. The same methods apply to Stamina and Health/Medicine "tonics" at 11:00, 12:00 and 1:00 on the radial menu.

[tadp0le]

It is meant to serve as an indicator. Most of the time it is down to your horse being tired or hungry, it may be injured as well. Moosh89 covered this perfectly. Be sure to give Moosh an "UP" to the left if you appreciated the answer  

Edited January 25, 2019 by tadp0le

[liquidfire]

It is just an indicator like mentioned. I guess this is the easiest way they thought to bring attention to care for your horse. Shove some grub in its mouth and see if it improved.