I think the story that works best, for me, is as follows.
The Air Ministry wished to assess 3-engine flying boats to either replace of suplement the Supermarine Southampton. Short were in the process of designing the Calcutta to meet an Imperial Airways requirement, similar in many ways to the Ministry's thoughts on a patrol flying boat. Unusually for a civil aircraft at this date they issued a specification around the Calcutta, which was 14/26, as they wished to assess the aircraft thoroughly. This was probably not put out to tender. It is known that the Calcutta spent more thime than was usual being assessed at Felixstowe before eventually being passed on to Imperial.
While the Calcutta was being constructed the Ministry issued Spec 4/27 for a 3-engine patrol flying boat and received tenders from Saunders, Blackburn and Supermarine. In an unusual move they decided to order all three; the Saunders Severn to 4/27 and then extra, but very similar specs. for the other two, 5/27 for the Blackburn Sydney and 6/27 for the Supermarine Southampton MkX. As we know Short produced military derivatives of the Calcutta while Blackburn built the Nile, a civil version of the Sydney. Supermarine planned several civil version of the MkX, mostly named Sea Hawk.
Avro may well have submitted the Types 606 and 607 for 4/27 but, as I said above, they look rather dated designs and may have originated earlier. They seem unlikely to have been suitable for 4/27. As with Saunder, Blackburn and Supermarine they no doubt had civil derivatives in mind and may have offered these to Imperial, but I doubt they were linked to 14/26 directly.
edit...One more thing. Supermarine tendered two designs to 4/27, a biplane and a monoplane. The biplane was revised extensively to become the Southampton MkX while the monoplane was used as the basis for the design of the Air Yacht