I like the film more than the book.
Here 'more' is an adverb, modifying the verb 'like'.
She earns a lot more than I do.
Here, 'a lot' (meaning *to a great extent/degree') is used as an adverb. Again, 'more' is also used as an adverb modifying another adverb 'a lot'. Similarly, in the expressions, 'a lot more', 'a bit more', 'more' is used as an adverb.
Reference : -
https://www.macmillandictionary.com/dictionary/british/more
https://www.grammar-monster.com/easily_confused/alot_a_lot_allot.htm
'More' is used as a determiner only when it is followed by a noun :
She earns more money from her business.
She spends more time with her boyfriend. etc.
Therefore,
'More' is a determiner when followed by a noun' :
'She earns more money.'
'More' is a pronoun when used instead of a noun :
'He did more for me.'
'More' is an adverb
(1) when used with a verb to modify it:
'She wishes to earn more.'
(2) when used before an adjective to qualify it :
'She is more beautiful than her sister.'
(3) when used before another adverb to modify it :
'She comes here more often.'