Cycling UK Forum

Around New Alresford there are some lovely little lanes and tracks, or there used to be when I lived there in the last century
The Watercress Line steam railway is great for children too

There is a reasonably flat route along the northern edge of the Southdowns along some quiet roads. Start at West Meon, then East Meon, Oxenbourne, Buriton. If you want to venture into West Sussex then continue to South Harting, Elsted, Didling, Cocking, Heyshott, Graffham. Also there is a good route along the Itchen river starting at New Alresford, then Ovington, Avington, Easton.
If you want dead flat then there is a cycle path all the way along the seafront from Tichfield Haven to Gosport, then get the Gosport ferry to Portsmouth, where you can pick up the cycle path along the seafront at Southsea, then if you want further nautical adventures you can get the Hayling Island ferry! Best to check the wind forecast before going on this route as it can get quite blowy and it's hard work going in the wrong direction.

jonorris72 wrote:. . . reasonably flat & preferably off road routes . . . .

The Meon Valley Trail is a former railway line that runs from Wickham up to West Meon - approx 11 miles.

The hardcore surface is mainly good enough for road bikes.

Wait until it has dried out for a couple / several weeks.

Be aware that there is a slight drop down, on ramps, to road level, @ Exton. This road crossing is often flooded with a shallow lake between ramps.

https://www.hants.gov.uk/rh/cycling/meon-valley.pdf

In the east of the county:

The Meon Valley Trail is a decent option. Wickham has a few places to eat, and The Shoe at Exton is also accessible via about a mile of quiet roads (you have to cross the A32 but it's in a 30 limit, it's not too bad).

Alice Holt (between Farnham and Bordon) is a reasonable place for kids, though not super flat. Often quite busy.

The Hayling Billy line is child-friendly, too, although fairly short.

And just over the Sussex border there's the Centurion Way from Fishbourne to West Dean.

In the west…

The Test Way is traffic-free from Timsbury to Fullerton, though I've never ridden it.

Not cycling but..
The Watercress Line organises a walk along the track, 10 miles on the last winter weekend before trains to start running
That is enough to get pleasantly tired

Have you considered that the problem might be the gearing on your daughter's bike? Maybe if she used lower gears, it would be easier for her?