Fundamentally, its a cultural issue we have in this country that is somewhat disgusting. Politicians should not be just listening to their voters, or their backers, they should be listening to all their constituents, everyone who lives in the areas they represent, with particular focus to the most vulnerable and not the other way around as what frequently happens. The incentives are completely backward. The interests of those that play the politick game well are prioritized at the cost of the people who need the most assistance from public government.
I'm very supportive of expanded voting and recently LA has made election day into an entire week that you can stroll in and cast your vote (Of course, everyone waited until the last minute and polls wrapped around the block at the very last day you could cast your ballot, but that is beside the point).
Even with vote by mail or other initiatives to get out the vote, this leaves a lot of people without representation in LA. There is the requirement that you have the time to study the issues and become an informed voter, which is a privilege not enjoyed by everyone. There are also a lot of undocumented people paying rent and working jobs in LA. I think these people who are contributing to the local economy should also be represented by the people in charge of the area in which they live their life. They have just as much right to be here as I or any other citizen does.
I think the best way to overcome petty politicking would be to limit the control elected officials have over what should be logical and factually rooted decision making. In LA, council members are more powerful in their district than the mayor or any other elected official, they have absolute control over what gets built be it on parcels of land or even paint on the roads. Just look at the patchyness of the bike lane network to see this effect in action; metro has money earmarked and is ready to build but local council members just refuse to allow it to happen in their district. They operate as little feudal lords who award contracts to friends and deny contracts if some who holds their ear takes issue for whatever reason at all.
Urban planning decisions should not be controlled by politicians, they should be controlled by urban planners who are highly trained civil engineers from the best engineering schools hired to do an apolitical job. They understand these issues better than anyone else in government, and their decisionmaking is rooted in the cutting edge theories and ideas present in their field, profit and personal preference be damned.
This is how public government should be run, deferring decision making to informed experts rather than the wills and wishes of those who command the most influence over their representative.