It can, and that meaning is very close to how "cover the costs" is used in English, and as in English, although it is not the primary meaning, it is very common. In general, anywhere you would use "cover" in English, you can use cubrir.
- News coverage -la cobertura de noticias
- Cover the debt-cubrir la deuda
- Insurance coverage-coberturas de seguro
- If you need a car, we have it covered-Si necesitas un coche, lo tenemos cubierto
One place where you will see some difference is when you intend that someone talked or wrote about some topic. In this case, English uses "cover" more often than Spanish uses cubrir. So while "We will cover that in our report"-cubriremos en nuestro informe makes sense, you will more often see other verbs indicating that "we will see", or "we will find" in that place.
Hydro, saying "pick up, firm up, get up, speed up" are all common times to add "up," but it is overused, and can be left off much of the time, like when one washes (up) the dishes - no need for it there, or when one shakes (up) a can of paint. You could "shake up" an organization or the office, though, and "cover up" a crime!
I think encubrir would be the correct word for cover up. Cover or cubrir could mean insurance coverage or the like. Encubrir or cover up would have to mean covering with a tarp, paint, toilet paper, etc. Examples:(After we toilet paper their cars you can cover up my car to make it look like we didn't do it) (Since I don't have money for insurance, you can cover my car. You use it more than I do anyhow.) Maybe a native could help us out on this one?