As it is currently wintertime, the short days are close at hand, so we should use "these" instead of "those". Some people may not have taken this paragraph to be about this winter in particular. In that case, "those" may be correct.
New Year is a proper noun, so it needs capital letters.
3. Quantifier error:
"Much" is used with uncountable nouns. Since "people" can be counted, use "many" instead.
4. Verb form:
For typical or repeated activity, use the simple present tense. People "enjoy" things. And since they enjoy things, we need a noun as the object. In this case, we can use the gerund: "resting" and "eating".
5. Word form error:
We need an adjective, not an adverb, before the noun, so "traditionally" should be changed to "traditional".
We often use a colon to introduce a list. In this case, "traditional foods" is followed by a list of examples, so a colon would be appropriate here. A dash ("—") could also be used; just don’t confuse it with a hyphen ("-").
7. Plural or singular:
When we are referring to types of food, "turkey" and "fruit cake" are not usually countable, so these nouns should be singular.
8. Article use:
"All the trimmings" is an idiom that takes the definite article ("the"). "Fruit cakes" and "plum puddings" may also use the definite article (to indicate that these are familiar kinds of foods), but the result is a little stiff.
9. Word form:
This is tricky. The correct word is "mincemeat". It sounds like "minced meat" but is quite different. Mincemeat is a mixture of fruit in a sweet sauce.
10. Gerund or infinitive:
"To sit" is grammatically correct, but it is more usual to say "sitting" in this context.
11. Noun or adjective:
"A benefit" is grammatically correct, but the adjective "beneficial" sounds more natural here.
12. Word form:
"Weathers" is not a plural; here the meaning is "weather is", which can be contracted to "weather's."
13. Word confusion:
When you are referring to a time, you need to use "then". The word "Than" is used in comparisons.
14. Verb tense:
This is similar to (4) above. When something is generally true, use the simple present tense ("like"), not the present progressive ("are liking").
"But" introduces an independent clause, so we should put a comma before it.
On these short winter days over the Christmas and New Year holiday, many people enjoy resting at home and eating traditional foods: turkey and all the trimmings, fruit cake, plum pudding, and mincemeat pies. Sitting by the fire is also beneficial as the weather’s usually cold then, though some people like participating in winter sport.