Internal Laying/Egg Peritonitis:
Distended bellies, especially in older hens with reduced laying, can be a sign of internal laying where eggs are laid internally and absorbed or cause inflammation (peritonitis). This can be serious and lead to infection and death.
could also be reproductive Issues. as hens age, their reproductive systems can become less efficient, leading to issues like egg binding, soft-shelled eggs, or internal laying, which can all cause abdominal distension and reduced laying.
What food/nutrient do you provide ?
Less common but possible causes for distended bellies could be tumors, ascites (fluid accumulation), or organ issues. These would likely require veterinary diagnosis
Nutrient Supplementation:
You can try supplementing their diet with a high-quality layer feed, calcium (like oyster shell), and potentially a vitamin supplement if you think they might have a nutritional deficiency. This may help improve their overall health and potentially egg production if the issue is mostly-- if not completely-- nutritional.
Pictures can also help- I could be completely wrong too !
It is your choice, but i'd recommend only culling the hen IF they seem overly uncomfortable and are suffering, and you are unable to help them.