I
prayed that he would recover from the depression from losing his father.
the key word here is the main verb "prayed". If we take a closer look at what follows this main verb, it is nothing but a phrase that indicates or is built on some form of hope. When it comes to hope, you cannot remain 100% certain of the outcome of that hope.
In this case, the word "will" represents certainty of achieving the desired outcome (In other words, the word "will" is used in situations where there is 100% certainty). Since the propmt is trying to convey hope in some form, the usage of the word "will" does not seem apt for the main purpose of this passage. Hence, we can eliminate (b) and (c) since they use the word "will"
A. prayed that he would recover from
B. prayed that he will recover fromC. prayed that he will have recovered fromD. prayed that he would recover off - the general idiomatic structure involving the word "recover" is "recover...from". This structure does not exist in (D).
E. prayed for his recovery of - the general idiomatic structure involving the word "recover" is "recover...from". This structure does not exist in (E).
Hence,
(A) is your answer choice.