In general, the answer is complicated, though it is similar, but not identical with English.
First, grammarians (at least the ones at Wiktionary) use 'determiner' for a class of words that includes articles (definite and indefinite) and words that can replace articles. For example Auto requires a determiner which can either be an article — das Auto, a possessive determiner (also called a possessive adjective) — unser Auto, an indefinite determiner, irgendein Auto, an interrogative determiner — welches Auto, a demonstrative determiner — dieses Auto, or a determiner in one or two additional categories that I can't think of at the moment, or perhaps an oddball that is difficult to classify. Adjectives, which can also come before nouns, do not replace an article, so das rote Auto, and so adjectives form a different word type. So the question should really be: Which nouns require a determiner and if so when?
Determiners that are not articles can always be added before a noun that does not need one. You can even add articles in front of a noun that does not need one, but it changes the meaning. For example
Ich mag Käse. — "I like cheese (in general)."
Ich mag den Käse. — "I like the cheese (at this restaurant)."
German nouns can be classified according to whether they require a determiner or not. Some do, some don't, and for some (as in your example Unterstützung) a determiner is optional. English and German usually agree on this, for example plurals and nouns for substances (for example Holz — "wood", Wasser — "water", Käse — "cheese") do not require a determiner. English and German don't agree on abstractions, in English they usually don't require a determiner but in German they usually do. For example die Liebe — "love", der Tod — "death". Names usually do not use articles in German (at least the standard variety; articles are used in the south), but when preceded by an adjective they do require an article. I won't attempt to enumerate every rule and exception here; many apply to specific situations which don't occur very often.
The issue of whether there is a preposition involved usually makes no difference. There, however, an exception to this in that a determiner is not needed with ohne, particularly when there are no adjectives or other modifiers attached. For example:
Er kann ohne Flugzeug fliegen. — "He can fly without a plane."
The sentence "He can fly without plane," would be ungrammatical in English.
