In the case where they are hoping to achieve a "majority", ... could also be interpreted to mean that they are hoping to at least win the most seats of any competing party even if this meant winning less than half and that they would be required to seek a coalition partner or to rule as a minority government. The latter interpretation being more likely in a situation where you have, for example, all the polls and pundits predicting that they've got no chance of winning outright (ie. more than half the seats) but are still in the race for a lesser win.
Apologies for all the toing and froing. Again, I emphasise I only brought it up as a possible and perhaps acceptable single-word term, in response to the OP.
I'm not advocating that it should be used instead/ahead of the other terms, of course. The point is, however, the term is being used widely - the abundance of google results clearly attest to that.
In the political context.
What you've described is a minority government, whereas the term is "majority" (which in all likelihood is short for/ synonymous with any or all of "governing majority", "absolute majority", etc., - take your pick).
We are not talking about a party merely winning the election, by gaining more seats relative to individual rival parties - that's not a big deal. The article would just read, "Syriza or party X hopes to win the election". Ho hum. In Syriza's case, that would have been aiming very low given the pre-election polls (I'm referring to Syriza as it's topical).
So the question with Syriza was, will it win the necessary number of seats that will allow it to govern in its own right, will it reach "majority"? Majority here is a status, a governing majority, a majority government. That's the crux of it.
That's what achieving/winning a majority means for political parties in elections, it's about reaching a majority government, this status, and that's how it's being used in newspapers etc. That's what it's all about, the ultimate prize is having outright control, not just winning the election.
So, the article writers have this array to choose from:
single-party/full/ absolute/governing/etc.majority , but they'll often just choose the single term "
majority" instead, which they obviously view as being synonymous with the above terms. They must expect their readers to readily understand it this way, too?
Maybe we need to check that this is indeed what they want to convey with the single term, now and again. But for the most part, I'm satisfied that it is.