Learn Hardcore Croatian: Ovaj tjedan imam težak zadatak.

Formally, Ovaj tjedan is an adverbial time expression in the accusative (masculine inanimate accusative looks the same as nominative: ovaj). You can also say Ovog(a) tjedna (genitive) with the meaning during this week. Both are common:

Ovaj tjedan radim puno. = This week I work a lot.
Ovog tjedna radim puno. = I’m working a lot during this week. Using U ovom tjednu is possible but less natural in everyday speech; Za ovaj tjedan means for this week (allocation/plan).

Why is there no article before težak zadatak? How would I say “a difficult task”?

Croatian has no articles. Težak zadatak already means a/one difficult task from context. If you want to stress the number one, you can add jedan:

Imam težak zadatak. = I have a difficult task.
Imam jedan težak zadatak. = I have one (single) difficult task.

Why is it težak zadatak and not teška/teško? How does agreement work?

Adjectives agree with the noun in gender, number, and case. Zadatak is masculine singular (and here accusative = nominative in form), so the adjective is masculine singular: težak.

Masculine: težak zadatak
Feminine: teška knjiga
Neuter: teško pitanje Note: Although it looks nominative, both the adjective and noun are actually in the accusative as the direct object of imam.

What does imam mean exactly, and how is it conjugated?