Is it true that despair and low resolve concerning supplication are signs of the imminence of relief, and that Satan knows this, so he whispers to the believer to abandon supplication? And can Satan know about the imminence of the answer?

Despair and a decline in resolve regarding supplication are not indicators of an imminent relief. There is no evidence that Satan knows about this and consequently discourages the servant from supplicating for this reason. However, Satan strives to divert the son of Adam from all good, including the remembrance of Allah and supplication. Allah Almighty says: "Satan only wants to cause animosity and hatred between you with intoxicants and gambling and to avert you from the remembrance of Allah and from prayer." Supplication is a form of remembrance, so a believer must increase in supplication and not slacken in asking Allah in every circumstance.