What is the OA??

I am not sure but I can give a try

Would encourage seems redundant to me

So I will go with D

X encourage Y to do Z

US encourage Turkey to Withdraw

Correct me if I am wrong

Karan

What is the underlined portion. I guess there is an error in your posting.

B. of the United States that it would encourage Turkey, which invaded the island in 1974 and now controls their

'of' not required, also 'their' is incorrect
C. that the United States would encourage Turkey, which invaded the island in 1974 and now controls its

not sure if 'that' is required.

D. That the United States encourage Turkey, which invaded the island in 1974 and now controls their
'their' is incorrect

E. the United States to encourage Turkey, which invaded the island in 1974 and now controls its right.
best option for me.

IMO A

The recently reelected president of cypres has asked that the united states would encourage Turkey, which invaded the island in 1974, and now controls their northern regions, to withdraw from his country.

President and his need to agree

I think the question's underlined portion & the options are wrongly typed.

Assuming: the underlined portion to be "that the united states would encourage Turkey, which invaded the island in 1974, and now controls their"

A: "...president... asked... US would encourage..." awkward and illogical, ASKING and WOULD do not go well together. "Would" implies commitment to an action, so president's asking the US and also making a commitment on US's behalf does not go all that well.
Ask -> Order or a command, hence should be followed by the infinitive form of the verb (To included, like to encourage, to submit, to call...)

B, C: same reasons as stated above.

D: ok, it removes the WOULD, but does not use "to encourage", also the pronoun error... "which invaded the island in 1974 and now controls their northern regions... " the pronoun their is referring to ISLAND, the correct pronoun to refer to the island is IT, as ISLAND is not a living thing.
Also, "asked the US" is more idiomatic than "asked that the US" since in this sentence, US is the direct object.

E: Has to be correct, right? but we need justification too
IMO E is correct because, 1. It uses the correct for of the verb "encourage", 2. It uses the correct pronoun for "Island".

Assumption: The word "RIGHT" appended to the end of option E is not a part of the option. It does not fit. Could it be that the poster was trying to indicate that E is the OA?

"asked" is not a reporting verb, hence "that" is unnecessary. Thus A,C,D are gone.
"asked of" - not idiomatic B is gone

E - Best - has only asked, maintains parallelism with "to" - Hence correct.

The sentence contains 2 errors:

1. "that the united states would encourage Turkey" => would is redundant
2. "now controls their northern" => It refers to a Singular term "island" but uses a plural form of preposition "their". The correct form would be to use "its"

BAsed on 2, B and D are out..
Out of C and E, C still uses the redundant structure by including "that... Would encourage ...." and moreover, E is concise and clear in meaning and is grammatically correct.

Although i think Statement E has a typo or something.. umaa, could you please verify and correct the errors..