The post-Hawkins cruiser's missing middle role

Was the 12,000-to-13,000-ton post-Hawkins cruiser intended to fill the gap between ordinary cruisers and battlecruisers?

Yes; it was meant to restore an intermediate capability as the armored cruisers that had previously occupied that layer disappeared.

The answer links the proposal to a force-structure gap made increasingly visible during the First World War. Tiger and the other battlecruisers sat well above ordinary cruisers, while the aging armored cruisers that had once filled part of the intermediate role were leaving service. A large post-Hawkins design was therefore not simply a cruiser with more guns and displacement. It was an attempt to rebuild a useful layer between the scouting cruiser force and the capital ships.