If you specifically mean the vessel has some liquid in, but not enough, and you are adding more liquid - the same sort, or a different one - to fill it to a required level, that's the phrasal verb to top up1.
The jug of milk was half full, so he took a glass of water and topped it up.
Phrased a bit more naturally:
The jug of milk was half full, so he topped it up with water.
(Topping it up with water sound a bit weird, but okay)
You get this in cocktail recipes:
Cut a lime into quarters and place two quarters into the glass. Add the gomme syrup (or sugar) and rum and muddle together, making sure you squeeze the juice out of the limes. Fill with crushed ice and top up with soda (or apple juice for an apple mojito). Add the sprigs of mint and tease to the bottom of the glass with a spoon, at the same time as gently lifting the lime, sugar and rum upwards.
(From The Guardian's beginner's guide to making cocktails)
You can also use that for materials other than liquids - you might talk about topping up your liquor cabinet, or your supply of pasta. It can even be for immaterial things, like topping up your mobile (cellular) phone's credit, or topping up your bank account by selling unwanted possessions. In these other contexts, it doesn't usually carry the connotation of filling to a specified level, it just means that you think there's not enough and you should get some more.
1: At least, in British English; American English apparently uses "top off", which we use... differently, in British English.
Best Answer
Although not exactly a single word, the phrasal verb to throw away is as specific as it can possibly get when describing a situation where you want to get rid of something that you no longer need. It strikes me as a little bit odd that you've never heard of this, rather basic, English expression before. I'm pretty sure you have, though. You just forgot.
Examples: