'Spanish archer' is a slang phrase.
El (Spanish word) + Bow (what an archer uses.)
i.e. elbow.
If you give someone the Spanish archer, you give them the elbow (which is itself a colloquial phrase) and it means to break up with someone. I fully understand your confusion, 'Spanish archer' is a phrase I've only ever heard used by people from London, and 'giving someone the elbow' is I believe a British phrase itself. I imagine many native speakers would also be confused by it.
The lyrics would seem to reference a break up, although perhaps a bit indirectly.