Perhaps a simple checklist will help you:
1. Is this a countable noun?
If it is uncountable, there is no need to think of using "a" or "an" in the first place, as they refer to one of a countable noun (e.g. a pen, an apple).
2. If it IS a countable noun, are you referring to it specifically?
If your answer is YES, you have to pay attention to the next part:
(a) Something that you are mentioning specifically may be unknown to the reader (perhaps you are introducing it for the first time). In this case, use "a".
A dog approached me. (You know which dog it is. It is a specific dog that you can see coming towards you. However, since this is the first time your readers are reading about it, use "a".)
(b) Something that you have already mentioned previously to the reader
A dog approached me, and I petted THE dog. (Here, the reader knows it is THE dog which approached you earlier.)
If your answer is NO, that means you are referring to any one out of all the same nouns. Use "a".
A plane can fly.
Back to your question, whether "a" should be used does not depend on the presence of compound adjectives. Instead, it depends on how specific you are in referring to a noun and if there is a shared experience of that noun with your readers.
A: I am flying to London on Wednesday to attend A two-week training programme.
[You know which programme you are attending. The reader, however, does not.]
B: I am flying to London on Wednesday to attend THE two-week training programme [that you attended last year.]
[You know which programme you are attending. The reader also knows exactly which one you are referring to because he/she attended the same programme before (shared experience).]
C: I like to fly to London to attend two-week training programmes.
[In this sentence, you do not even need an article, because you are making a general statement (like a personal habit) without referring to any particular programme.]
I hope this helps.