- I don't have any special plans for this weekend.
Q1: What is the meaning of 'for' here? Q2: What is the part of speech of 'for this weekend'? Does it modify 'any special plans'? Or does it modify 'don't have'?
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In OP's context, including the preposition strongly implies that for this weekend should be interpreted as an "adjectival" element modifying the plural noun [special] plans.
Without the preposition, this weekend is an "adverbial" element modifying the verb [to] have [plans].
In practice, it doesn't really make any difference to the overall meaning in the exact cited context, but there could be a difference in, say,...
1: I paid the bill for last week
2: I paid the bill last week
...where exactly the same syntactic distinction applies.
In #1 it's the bill that relates to [goods / services delivered] last week - which might be paid for later.
In #2, it's the act of paying that happened last week, where last week adverbially modifies the verb-based action [to] pay [the bill].
It may help to note that it's not possible to say I am paying the bill last week, but it is perfectly natural to say I am paying last week's bill or I am paying the bill for last week.