Translation:The American director has a lot of money.
It is something like "her money is plentiful". Yes, I guess it should be covered somewhere in the Tips and Notes. This is a common structure.
Actually, if we dig deep into it, this is nothing more than a possessor-possessed structure, where the possessed is a predicate-subject structure. To break it down:
"Kevés a pénz." - Or "a pénz kevés". - The money is little. It works either way.
"Pénz" is the subject.
"Kevés" is the predicate.
Place it as a possessed into the P-P structure:
"A rendezőNEK kevés a pénzE."
I guess we are forced to "spell out" the possessor-possessed suffixes, because of the inserted predicate ("kevés") in-between. But we can turn it around and the possessor suffix will become optional:
"A rendezőNEK a pénzE kevés."
"A rendező pénze kevés."
So, in either case, "kevés" is the predicate in these sentences. That is what we are stating about the money.
If we change the role of "kevés", and put it in an adjective role, then we need to introduce the verb "van".
"Kevés pénz van."
"A rendezőnek kevés pénze van."
Sok pénze van is also correct. Regarding to your question.
He has plenty of what? Of the money . A pénzből.
So the definite article "a" can be used before a noun in this case.
He has a lot of money. - Sok a pénze/Sok pénze van.
He is lack of money. - Kevés a pénze/Kevés pénze van.
Hope this helps.