can you name any of their competitors in the second mentioned league?
Many English clubs plus clubs from Wales like Cardiff City, Swansea City and Newport County.
English Football League
Although primarily a competition for English clubs, clubs from Wales – currently Cardiff City, Swansea City and Newport County  – also take part, while in the past Wrexham, Merthyr Town and Aberdare Athletic have been members.
how many premier league titles does manchester city have UNANSWERABLE what type of leagues do they participate in? Premier League, Football League
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English Football League [SEP] Introduction
As this sponsor changed over the years the league too has been known by various names. Starting with the 2016–17 season, the league has moved away from having a title sponsor, re-branding itself as the English Football League (EFL), in much the same way the Premier League is known as the "EPL" internationally.