Can I add tomato paste afterwards, instead of during cooking, for a tomato sauce? So I have a pizza tomato sauce recipe that called for adding 6 tablespoons of tomato paste during the cooking. Unfortunately, due to being very disorganized, I only realized half way through cooking that I didn't have any tomato paste in the house and the local 24h corner shop didn't stock any.

I carried on cooking the tomato sauce topping without adding the tomato paste and then put it in the fridge overnight. Should I pick up some tomato paste and add the "finished" product to the frying pan, heat up and put in the tomato paste or should I just leave it. Does the tomato paste add anything bar depth of flavour?
In general you need to consider the purpose of the ingredient, and what would happen to it during the cooking that it's missed.

For example, there's no point stirring a spoonful of flour into a sauce after cooking, since the flour needs heat in order to thicken the sauce.

In the case of the tomato paste, not much will happen to that particular ingredient during cooking. It may have caramelised a little, depending on the cooking method. It would have infused more deeply into some of the other ingredients. But these are subtle things. Taste your finished product. If you feel it would benefit from an extra boost of tomato flavour, by all means stir in some tomato paste.