Blocked up is treated as an adjective.

Sometimes you will see participles (or gerunds) of verbs used as adjectives. In the phrase "a blocked up sink," the words "blocked up" are functioning as an adjective, therefore you see article + adjective + noun (a + blocked up + sink). If you were to turn the sentence around and use "block up" as a verb, you could say that "Hair and food can block up a sink." Note, while "up" is usually a preposition, in this sense it is a verbal adjunct. "Block up" should be considered as a unit to be the verb. This is a remnant from English's Germanic origins. You will hear people say "Hair and food can block a sink up." According to prescriptive English grammar, this is incorrect as you should not end a sentence with a preposition (even though in this instance "up" is a verbal adjunct).

"blocked up" here is used as an adjective, to describe the sink. So in case of an adjective before a noun, the article is placed before the adjective. Example: a hot stove.

It should be written "blocked-up." That makes it clear that "blocked-up" is an adjective that describes the sink. You could say something "caused a blocked-up sink" but you could also say something "blocked a sink." That would probably be the better way to write it. "Can" can only be followed by an infinitive. You could say "Hair can block up a sink" because "block" is not conjugated. You could not say "Hair can blocked up a sink" because the blocked is conjugated, and therefore that sentence does not make any sense. I hope this helped.

"can cause a block up"