a) In North America a "precinct" is a district of a city. In the USA a "Democrat worker" is a person who works for the Democratic political party. So a Democratic Precinct Worker is someone who works for the Democratic Party in a specific district of a city. b) a limp is any kind of impeded walking. If your foot is broken, you will walk with a limp. c) lore is the collection of traditions and knowledge on a subject which is held by a particular group. "family lore" is the traditions and knowledge about a particular family d) when you "beckon" someone you are making a gesture with your hand to encourage them to approach you. You can use this word in other situations "her smile beckoned" means a person saw her smile and felt compelled to approach her. "the sea beckoned" means a person felt an urge to go sailing on the sea. "Decent public schools beckoned" means that because they were decent (ie: good) that people wanted to go to them.

b: To Walk with 'a limp' means that you have something wrong with one of your legs or feet. In medical parlance we talk about 'short stance phase' which means when walking, one foot contacts the ground for longer than the other foot. This is usually because one foot , hip or knee is painful. c: 'lore' means stories, history, legends as in 'folklore' - stories passed down (by mouth usually) from generation to generation. d: To 'beckon' is to entice; but this word is, again, only used in a literary context usually. The meaning here is that the public schools appeared attractive to the narrator.

In the US, voting areas are divided into "precincts". The Democratic party is a major political party. A Democratic precinct worker would have the responsibility of encouraging people in his precinct to vote for Democrats.