Is this sentence correct? "A lot of sugar have been added to the milk." Please explain it.
A lot of sugar have/has been added to the milk.
No, the verb should be the singular "has".
The quantificational noun "lot" is number-transparent, which means that the whole noun phrase takes on the number of the noun that is complement of the preposition "of", which in this case is the non-count "sugar".
Since non-count nouns like "sugar" take singular verb agreement it follows that the verb must be the singular "has".
No, sugar is an uncountable noun, and A lot does not quantify it, so it takes has: A lot of sugar has been added to the milk.
However, if you quantify it, you may say: Two cups of sugar have been added to the milk.
Although I think that BillJ's answer is probably a correct way to look at this issue, it seems a little technical. It's possible to look at this sentence another way and understand it with fairly basic, high-school level grammar.
The subject of this sentence is "a lot", which is singular. Therefore, the verb should be "has".
A lot has been added to my milk.
"Of sugar" is an adjectival prepositional phrase, modifying "a lot". It's telling you what kind of lot has been added.
Using 'lot' in this sense is somewhat idiomatic, in that a 'lot' is
an article or set of articles for sale at an auction. (MW)
Although it's become so common to use it in this fashion that the idiomatic usage is probably more common than the original one. It has this definition:
a considerable quantity or extent (MW)
A considerable quantity has been added to my milk.
What kind of considerable quantity?
A considerable quantity of sugar.
Regardless of how you look at it, you should use "has". It's an informal usage anyway, so the exact technical details aren't that important.
You could use "have" with the countable word "sugars" or similarly "sugar crystals". In the context of tea or coffee, you may ask for 3 sugars, where it is implied you are talking about specific units of quantity like cubes or packets.