Itinerary question

Will you have a car and do you want to spend 4 days in one place or are you happy to move around?

Since you’ll be in Hay on Wye, I would try to visit Hereford Cathedral to see the Mappa Mundi. Hereford is a small town (although technically a city??) but if you are into old documents they don’t come much older than this. Beware the local cider , especially anything called scrumpy.😀

Hay-on-Wye has wonderful countryside for miles around in all directions. The small city of Hereford has already been mentioned as worthy of a visit. Going NE a relatively short distance from Hay are some wonderful ‘black & white’ villages and you may like to continue in that direction to Leominster and north to Ludlow. If you do reach Ludlow, I suggest you drive about 1 mile north to Stokesay Castle (a medieval fortified house).

You will see a minor road going south from Hay signposted Capel-Y-Ffin / Llanthony. This road goes over the Gospel Pass in the Black Mountains. Whilst it is reasonable enough almost to the top of the pass, be warned that beyond it becomes one of the most difficult roads to drive in southern Britain. (You must be able to drive backwards in the event of meeting someone coming the other way).

SW of Hay is the Brecon Beacons National Park which the nutty Welsh Government have decreed should only be known by its Welsh name. This is some of the finest scenery in southern Britain and deserves at least a full day.

Going north westward from Hay, going A470 and follow the Wye Valley into the middle of Wales. On reaching Rhayader - go west to Elan Valley:> https://elanvalley.org.uk

It is possible to take a *back road over the Cambrian Mountains to Devil’s Bridge and perhaps return on the A44 to Llangurig and down to Rhayader and back to Hay. * You must be a confident driver.

So, plenty of interesting places to reach on day trips from Hay - provided you have a car.

I hope that you have secured somewhere to stay in or near Hay. If not do that immediately. It’s a small town with huge demand at book festival time. We resorted to glamping the last time we were there.

As you are a Librarian, and I assume will have a car, you could go on from Hay on Wye to North Wales, using Gladstone's Library at Hawarden as your residential base.
I'm certain it would work by public transport as well, but easier by car.
It ticks most of your boxes, except maybe Neolithic (but we can get that in the area if you want). The small cities are Chester (small ish) and St Asaph (it is a city, if a very small one).
Pretty much on the way there is also Brecon, another diminutive city.

Lovely place though it is, Brecon is a town, not a city.

John, Brecon has a Cathedral (Church in Wales) so is a city.

All of the above sound very nice, but if you're willing to travel a few hours after your visit to Hay-on-Wye, have you considered Chester? I was there a couple of years ago and just loved the vibe of it. It's a charming place with a lot to see and do, but still has a relatively small city atmosphere, with a lovely canal running through the city, some nice cafes, and just a lot of charm in general. I know there are some Neolithic sites nearby, castles, gardens and, of course, pubs.

One other advantage is that you could take a day trip into nearby Liverpool from there, which is a city that doesn't get a lot of attention but is really a lot of fun to see. It has some gorgeous architecture and a lot of interesting things in general.

Chester is north, not south of Hay-on-Wye.

On the librarians front, and not for this trip as it's too far out of the way, you might be interested on a subsequent trip in the Lit and Phil (Literary and Philosophical Society) at Newcastle upon Tyne.
I called in there this evening when passing after going to Evensong at the nearby Cathedral- the sign said that anyone could visit the Library upstairs, so I did.
A society with a very illustrious history.
One of those pretty special libraries, and they hold a lot of public lectures (many free) there as well. They were actually setting up for a book launch when I was there. I was very tempted but needed to get back to Carlisle.

Like many of the ancient Cathedrals Chester has its own library, which you visit by appointment, which would be very interesting to visit, even if they lost their very ancient documents at the dissolution.
Much closer to Hay-on-Wye there is the chained library at Hereford Cathedral.

Stuart - having a cathedral does not automatically make a place a city. For a specific example close to me Blackburn has an Anglican cathedral but is a town. Next door Preston is a city but does not have a cathedral.

No sure how far you wish to travel for your fours days but there are two castles you can actually stay in one is Bodelwyddan in North Wales but is really a folly castle owned by Warner's Holiday company. The other is Leeds Castle in Kent but obviously a lot further to go.

The nearest castles of note to Hay are:> Raglan, Chepstow (can be combined with Tintern Abbey), Powis Castle near Welshpool - which also has gardens. Medieval Tretower Court is also worth a look and about 4 minutes walk around the corner is Tretower Caste - but only a loan tower remains.