This one is tricky. It refers to a winning party (the party with the highest/greater number of representatives ["mayoría"]) that does not have a majority of overall seats (mayoría "en minoría"; mayoría minoritaria). This does not make sense in a two-parties "game" like in the US but it does in Europe. In England they talk about a hung parliament when this happens. You could use minority government/parliament. As the Asker states, a minority majority does not work for it has a fully different meaning; mayoría minoritaria refers to "a winning party without overall majority". Although I am not sure if there is a more succint expression to convey this in English.

In the particular context of the Asker, sus mayorías minoritarias estaban más cerca de la mitad más uno que en anteriores legislaturas; their minority government/parliament was closer to [a lean majority] than in previous parliamentary terms

--------------------------------------------------
Note added at 2 days 3 hrs (2020-11-06 22:12:56 GMT)
--------------------------------------------------

This may be it;

A plurality vote (in the United States) or relative majority (in the United Kingdom and Commonwealth) describes the circumstance when a candidate or proposition polls more votes than any other but does not receive more than half of all votes cast.
https://en.m.wikipedia.org/wiki/Plurality_(voting)#:~:text=A...



--------------------------------------------------
Note added at 2 days 3 hrs (2020-11-06 22:16:13 GMT)
--------------------------------------------------

i.e.
... their relative majorities were closer to [lean majorities]...

--------------------------------------------------
Note added at 4 days (2020-11-09 10:35:38 GMT) Post-grading
--------------------------------------------------

Hi Justin, I admit it is confusing and yet I still feel "simple majority" is not right in your context.

For example, on a yes/nay vote, absolute majority implies one option holds more than half of the votes of eligible members while simple majority conveys one option holds more than half of actual votes -disregarding absentees. See here for instance (US): https://www.google.com/url?sa=t&source=web&rct=j&url=https:/...

Also (US), on simple majority rule:
https://study.com/academy/lesson/simple-majority-definition-...

(Australia):
This section clearly refers to a simple majority, that is, a majority (half plus one) of the senators present and voting. A simple majority is distinguished from an absolute majority in the Constitution by the requirement in section 128 (...)
https://www.aph.gov.au/About_Parliament/Senate/Powers_practi...
(UK):
So there are 639 lawmakers who are considered to be active. That means any government needs 320 votes for a simple majority.
https://www.google.com/amp/s/uk.mobile.reuters.com/article/a...


Now, when more than two candidates are racing, simple majority voting determines the winner will be the candidate with the greatest number of votes.

Simple Majority Voting Issues
For instance, among a pool of four candidates nominated for a single seat, a winner can be determined from who gets the maximum number of votes. 
https://www.meridiaars.com/sometimes-a-simple-majority-isnt-...


But even here, they refrain from explicitly calling the greatest number of votes a "simple majority". It is only in Indian-English texts and some other translated texts that I have been able to find "simple majority" used with this meaning -other than some online-dictionaries definitions-, while "relative majority" is indeed used this way;


With simple plurality, the winner is the candidate supported by only a relative majority, that is, by a higher number of voters than any other candidate but not requiring any particular number, proportion, or threshold of votes. In practice, this makes it possible for generally binding decisions presumably decided by “majority” to actually be won by only a minority of voters. Plurality rule has traditionally been used in England and the United Kingdom and in modern times in former British colonies, including the United States, Canada, and India. Plurality-based electoral systems are also called first-past-the-post and winner-takes-all systems
https://www.google.com/url?sa=t&source=web&rct=j&url=http://...

And so this is why I wonder whether "simple majority" could be actually used in your context with that pressumed "secondary" meaning of "simple majority", for I feel it does not make full sense used this way here.

I am sorry anyhow for, as I said before, I might well be misleading you on this, complicating things unnecessarily. What I am certain about is what the Spanish original conveys, and that is not an actual majority (more than half of the votes); in English it seems "simple majority" could be used to convey this meaning of a less-than-half-of-the-votes majority -as per some dictionary definitions- but having found not a single example where it is used as such, well, I feel like Thomas: bring hither thy hand, and put it into my side. Again, sorry about that ;)