The most common and most likely answer to this crossword clue is the 7 letter word PRIMARY
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Buddy explains!

The straight part
We know that "Cardinal" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case cardinal = primary.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does PRIMARY mean?
PRIMARY noun- A primary election; a preliminary election to select a political candidate of a political party.
- The first year of grade school.
- A base or fundamental component; something that is irreducible.
- The most massive component of a gravitationally bound system, such as a planet in relation to its satellites.
- A primary school.
- Any flight feather attached to the manus (hand) of a bird.
- A primary colour.
- Primary site of disease; original location or source of the disease.
- A directly driven inductive coil, as in a transformer or induction motor that is magnetically coupled to a secondary
- To challenge (an incumbent sitting politician) for their political party's endorsement to run for re-election, through running a challenger campaign in a primary election, especially one that is more ideologically extreme.
- To take part in a primary election.
- First or earliest in a group or series.
- Main; principal; chief; placed ahead of others.
- Earliest formed; fundamental.
- Illustrating, possessing, or characterized by, some quality or property in the first degree; having undergone the first stage of substitution or replacement.
- Relating to the place where a disorder or disease started to occur.
- Relating to day-to-day care provided by health professionals such as nurses, general practitioners, dentists etc.