Are there days with free admission to the museums of the Circuit?

Yes, all museums offer some promotional or free admission dates that vary from museum to museum and year to year. In general, state-owned museums are free on the first Sunday of the month. The state-owned museums participating in the circuit are: Galleria degli Uffizi - Galleria dell'Accademia - Museo Archeologico Nazionale di Firenze - Palazzo Pitti e Giardino di Boboli - Museo Nazionale del Bargello - Cappelle Medicee - Palazzo Davanzati - Museo di San Marco - Opificio delle Pietre Dure - Museo Casa Martelli - Giardino di Villa Bardini. 
You are advised to always check reductions and free admission on the museum’s official website.