If someone wants to enter the masjid where people pray to listen to Dua Tawasul or dua in general that the Qari’ is reading, could they enter if they are in the state of haydh?

If it’s an official masjid, it’s not allowed to enter during the state of Haydh. One can sit outside of the masjid and listen. But if it’s just a prayer hall and not an official masjid then it’s ok.