Slow charging is generally a result of a poor condition or cheap cable. Like mentioned it could be the pins on the connector, always look at them before you plug a phone into them. If you see a burned/black pin discard the cable. Also like mentioned the last 20% will intentionally charge slow to reduce the strain on the battery which is because as a battery is charged and its voltage increases its internal resistance increases if a high current is maintained it makes heat. And since the battery is the resistance it would be heat in the battery.
For example this is a 12 watt iPad charger connected to a USB load simulator via a brand new Apple cable.
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Just to get it close to 12watt I needed to increase the load (thus current) to the point the voltage was dropping (due to the resistance of the cable) passed the safe 5volt +/-5% that a LiPo charging circuit would keep it. And thats a brand new brand name cable. Using a 10 foot long knock off cable from the local gas station....your not doing yourself any favors.
Here is a 6" cable. It gets a lot closer but using a USB tester plus the load simulator lets you see of the voltage, current and wattage drop through the cable.
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So yeah, if your phone is charging slow and the cable looks like crap then toss it.
Personally I wouldn't concern yourself with battery health. Obviously don't do something dumb like charge your phone on the dashboard of a parked car in the middle of the summer but they really aren't that expensive to replace anyway. Do we carry our shoes to prevent wearing out the soles?