Refers to the governing body of a specific locality, such as a city, town, or prefecture.
The local government is planning the construction of a new park.
With support from the local government, the shopping district was revived.
地方自治体 is a broader term for any local public entity, while 地元政府 emphasizes the governing aspect of a specific local area.
地方政府 is a more formal term for 'local government' as an institution, often used in political or administrative contexts, whereas 地元政府 carries a nuance of 'one's own local government'.
Compound of 地元 (じもと, 'local area') and 政府 (せいふ, 'government').