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".