John has three cups and (a) jug. Jane gave them the notebook and (the) pen
Can the article be omitted in these cases?
If we mean a countable form of 'jug', such as a container for a liquid, then we need the article. The reason is that if we omit it, the noun becomes uncountable by default. It does not matter whether uncountable 'jug' exists or not in any dictionary, the mere mention of it creates it for this text.
John has three cups and a jug
We have a different situation for 'pen', because we a list with a repeated structure:
Jane gave them (the book) and (the pen).
We can simplify this to:
Jane gave them the (book and pen).