Yes Lecce closes up in the afternoon, so unless you are super early risers and want to do a morning visit then a long lunch, plan your day trip for early evening when people come out, shops open, etc. You might spend the morning at the beach, have lunch, then get to Lecce and get parked, etc. around 4 so you can walk around and observe as the city comes alive.
There is a really interesting museum small enough that you can easily manage with the little ones:
https://www.nytimes.com/2015/04/15/w...ix-toilet.html

I am truly grateful to have stayed in Lecce so I could see it during the day and roam around at night -- seeing the ways that the differences in lighting played of its unique architectural details was fascinating.

We stayed in Lecce for four nights and Otranto for three nights, and this gave us more than enough time to see both places without being rushed, have beach time just outside Otranto, and take day trips to Gallipoli, Alberobello and the Grotte di Castellana.

We stayed in four different locations (+ in Matera), but we were in Puglia for 3 weeks.
With only one week I would stay in the Valle d'Itria (Ostuni or Fasano) and in the Salento (Lecce or Otranto).
Those are good bases for daytrips in Puglia.

I see you have switched locations, but since I am a Puglia fan, I thought I'd send in remarks.

I believe that kids should go abroad as soon as possible, with the complete understanding that they will NOT remember where they went, only what they ate and how much fun they had and that just because people speak another language does not make them "different". And it's important that we as parents in the early years get to enjoy the time we spent abroad WITH our kids, using these moments as our both current and retrospective reward for raising them.

With that in mind, here are are my thoughts...

• With the kids, Matera has to go off the list. We took our kids on hiking vacations in the Rockies when the youngest was 7 or 8, and we had to make each kid in charge every day of the "Are we having fun yet?" barometer because it can get tough. Matera's terrain would defy that mission: it is unforgivable with not that much kid appeal (my kids were more into art). Yeah, it sort of looks like a wondrous park in Disney World--but with an obstacle course, and Mickey ain't going to show up any time soon.
  
• I think a night and one day in Lecce would be a kid's paradise, especially if you go off of any sugar restrictions for the time period. I don't like dessert, but I ate every possible sweet and pastry in Lecce. Bet you the kiddies would remember what they ate there until adulthood. Strollers work well in the town.
  
• Any other Italian location or hotel, as long the hotel or masseria has a pool and is near a nightly not-too-crowded passeggiata, will work for your kids. For example, Bari looked like an ugly ghost town when we arrived mid-afternoon. By 8 pm, every kiddie in the world was out on the town with their parents for the passeggiata, doing fun "bad" things kids like to do. GREAT!

Happy planning!
AZ

Thanks everyone for the advice especially regarding lecce. I am now leaning more on our second base being in Lecce (or next to it e.g. in Torre del Orso as suggested) or in Otrano. Our travel dates are in flux now but hopefully I will be able to confirm them soon. Then I can go ahead and make my reservations =)

Haven't switched locations. Still only visiting Puglia =) Thank you for the wonderful advice. We decided against Matera . Like you and many others said, it is probably too much for the kids. Hopefully when they are older!

I'm going for 12 full days at the end of April. I've researched more places than I will probably be able to visit but I can see going back.

I did plan 3 days in Bari, in order to visit some of the places in the upper part of Puglia. I found that Matera is an hour's drive, so I'm going to try to make it out there.

But I'm trying to squeeze in some days in Gargano, specifically Vieste, which seems to be off the radars of many people visiting Puglia. Is is quite distant from the rest of Puglia so I hope it's worth going up there.

Lecce does seem like the best location for Salento, since you drive about an hour in each directions to reach many of the best-known destinations down there.

The Gargano was on y own radar for a long time but the more I investigated, the further down my list it went. Vieste is supposed to be great, but much of the rest of the peninsula, from what I have read, has been overdeveloped to aa ruinous degree. There was someone here who visited, I believe. I'd love to hear about your trip if you do get there..and anywhere else for that matter.