Could entrained air flow could be used to increase thrust in the similar way to how a bypass fan works? - The total thrust is √(2𝜈𝑃) where 𝜈 is the total exhaust effective mass flow straight backwards, and 𝑃 is the power input, from fuel. If the mass rate can be increased as with a bypass fan, the thrust can increased. This could significantly reduce fuel required in early stage, where fuel use is greatest. The effective exhaust speed is reduced, but the increase in mass flow more than compensates.
It's called an air-augmented rocket.
They add a lot of weight, but yes, they work.
They've been discussed on SE here: Limits of bypass ratio in air-augmented (ejector-jet/ducted) rocket
At least, the investors of the Skylon project seem to think it's a good idea. Time will tell whether they were brilliant visionaries or rich people lighting a massive bonfire with money.
Not quite an "air-breathing rocket" but perhaps of interest anyway is the Star Raker design by Rockwell. It's relevant because the idea of using the atmosphere as a free reaction mass + oxidizer is the key working concept, IMO. Star Raker just put the "jet" and "rocket" portions of operations into separate components, rather than trying to combine them like Skylon does.
While the idea is attractive on paper, adding significant complexity to a rocket isn’t good for reliability, although just making the rocket bigger is straightforward within some practical limits. In the context is a rocket launch, the cost of a couple of tonnes of extra fuel is probably peanuts.