The 19 year old attacker joined Soham in the summer from neighbours Newmarket Town, and has made twelve appearances, the majority coming as a substitute.
Conway signs for the Thurlow Nunn Premier Division side with the purpose of getting some regular game time, and made a scoring debut as Borough drew 2-2 away to Great Yarmouth Town on Saturday.
Another player to leave the Lane is Aundre Wholas-Spencer.
The attacker signed for the Greens in November from Clapton, however after one appearance has joined Essex Senior League side Enfield 1893.