As far as I am aware, care mistakes do not carry over.
What is defined as a care mistake is the following process:
let either the hunger or happy meter completely empty (both being at 0% at the same time is risky, so if you want a high care miss character, make sure at least one of them has at least a slight amount filled in). Next, your tama will make a noise if the sound is on and say your name in a speech bubble; the bottom right icon, "attention," will be lit up. In order to get a care miss, you will want to ignore the tama for 15 minutes until the attention light goes away, then you can care for it and refill the bars either partially or completely, depending on whether you need to get another care miss for the desired character.
edit for better accuracy: I am doing care misses on my P's today; it takes 15 mins from the time the alarm sounds before the attention icon goes away.