CathInfo.com

As far as the sighs go, the one I know has SIGNIFICANT health problems, so that could be part of it.

Whether voluntary or involuntary, breathing abnormalities that could be interpreted as mere sighs could be caused by any of these:

•   Congestive Obstructive Pulmonary Disease (better known by initials COPD), which I suppose is most common in the U.S.A. as the temporal reward for decades of heavy smoking.  Perhaps only housekeepers, sacristans, trusted "altar boys", and long-time male parishoners would even be aware of such a habit, customarily confined to priests' residences or cells &c.  Comments would be needed from, e.g., Matthew or Ladislaus, to inform readers on the extent to which the polluting habit existed within their experiences in seminaries.

•   Shortness of breath from acute anemia, which can appear in senior years as an incurable blood disease that's unrelated to any habit of smoking (or not).  Altho' "heav[ing] deep sighs before each sentence in the mass",  if heard also in sermons or normal conversation, would be a symptom waaay into the life-threatening range.

•   (Anything else? [×])

I'd not be surprised if local fire marshalls refused to allow use of otherwise common prescribed oxygen gas (O₂) in the sanctuary, where altar candles or charcoal lit for incense could be considered an unacceptable fire hazard.  So enduring a worsening shortage of O₂ during Mass might be a priest's private cross to bear in return for continuing to celebrate Mass.

-------
Note ×: I suppose many maladies or infirmities could be added to this list, because I have no formal training in medicine.