Is it permissible to wear a shirt with the word "Al-Aqsa" written on it, understood to mean "Al-Aqsa Mosque," knowing that it may be exposed to sweat, and does this constitute a desecration of the glorious word (Allah's name)?

There is no impediment to wearing the aforementioned shirt, as it does not indicate any degradation of Al-Aqsa Mosque. The word "Al-Aqsa" is a term shared between the Quran and human speech, and it is not considered part of the Quran unless there is evidence to indicate that. As long as the wearer of the shirt does not intend to insult the name, there is no harm in wearing it.