31:19a As for the priests, the descendants of Aaron, who lived on the farmlands around each of their cities or in any other city,

There were priests who had Aaron as their ancestor, and who lived on farms around the priestly towns and any other towns.
-or-
There were descendants of Aaron who were sacrificers, but who lived outside the towns of priests and the other towns.

31:19b men were designated by name to distribute a portion to every male among the priests and to every Levite listed by the genealogies.

⌊The king⌋ appointed men to distribute the gifts to them as well. ⌊They distributed the gifts⌋ to every man whose name was found in the list of priestly families.
-or-
Certain men were named to share the food portions with them. They shared the gifts with every man and boy ⌊identified⌋ in the records ⌊of the clan of Levi⌋.

31:19a–b (reordered)

Hezekiah appointed men to take portions of food to the Levites whose homes were in the pastureland near the towns of the priests and near any other town. They distributed portions to them and to all the Levites listed in the official records.

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