Pick up that mess.

“Pick up that mess.”
What is the meaning of this sentence?
Does ‘mess’ here refers to left-over food?

What is the context?

A mother is asking her child in her house, “Pick up that mess.”
This is the context.
Does it mean that she should pick up the miss arranged things and put them in order?
She should pick up the left over food to waste bin and clean the table?
Please comment.

It could mean anything that could be described as a “mess” (untidy, disorganised, dirty, etc.).

It could be as you suggest, that it refers to the child’s disorganised or scattered possessions (clothes, toys etc.).

It is not possible to be sure.

When we are taking food, can one ask the other, 'Don’t use you messy hand to serve others." since that man is eating food by his hand and not using fork.
Please comment.

It’s possible I suppose, but in practice “hands” seems more likely.

Note also typo; should be “your messy hand(s)”.

“After taking food, please clean the messy plates.”
Is this sentence OK?
How is it expressed colloquially?

Use of ‘taking food’ is very formal whereas ‘messy plates’ is very informal, so the sentence is a strange mix of styles.
Formal - After taking a meal, please wash/clean the crockery.
Informal - After eating, please wash/clean the messy plates.

If you don’t mean ‘wash the plates’, but just want to convey the idea of removing them from the table, then it would be better to use ‘clear away the plates’ than ‘clean the plates’.
You could also say ‘please clear the table’.