I have three girls named Carr in my records for the Mordecai school:
Esther Carr of Pitt County, NC, attended the school from early 1813 to mid 1814 (so, three sessions). Elias Carr was the adult associated with her account.
Mary Carr attended the school in early 1815, and then for all of 1817 (so, again, three sessions, but these weren't consecutive).
Winnefred Carr attended the school for two sessions, early 1814 and early 1815 (two non-consecutive sessions). She was also from Pitt County with Elias Carr as the adult on her account.
Elias Carr appears in several places in the school ledger, including a mention in January 1814 of him paying tuition for his "daughters"--so that establishes the relationship between Elias, Esther, and Winnefred Carr, at minimum. He also appears making a payment in January 1817, so he may also be the parent or relative of Mary Carr. (But he's in the ledger again in April 1818, when none of the girls above was enrolled, so I won't assume that. Still, it's a start.) He's clearly not this Elias Carr, though there must be a relationship there, and thus to the girls at the Mordecai school.
And there is. The Governor Elias Carr had among his aunties a Winnefred Carr and an Esther Johnston Carr, and yes, a Mary Carr. Their father was Elias Carr (1775-1822) and their mother was Cecelia (or Celia) Johnston. (Winnefred was named for Elias's mother, and Esther was named for Cecelia's mother.)
Esther Johnston Carr (1798-1864), then, was 15 when she arrived as a student at the Mordecai school, and she stayed till she was 16 in 1814. In 1816, she married a local man, Allen Blount (1789-1828), at her father's house. The couple had six children between 1817 and 1827, five boys and a girl. Esther was widowed in her early 30s, and died in 1864, in the same county where she was born. Here's her tombstone.
Winnefred "Winnie" Williams Carr (1800-1855) followed her older sister to the Mordecai school in 1814, when she was 13. In 1822 she married a Dr. John Thomas Eason (1796-1864). They had ten children together, born between 1822 and 1843. The couple moved to Sumter County, Alabama, before the last two children were born (so, probably in the late 1830s), and Winnefred died there at age 54, having survived at least two of her children (William died at age 17 in 1842, and Elizabeth died at age 20 in 1853).
Mary Carr (1802-1822?) followed her two older sisters to the Mordecai school in 1815, overlapping with Winnefred for one session. She was about 13 when she arrived there. and 15 when she left. In early 1822, she married Josiah E. Fowle (1791-1822?) of Massachusetts, but apparently (according to family tradition) faced her father's "violent opposition" to the marriage, and was disinherited. The story continues that Josiah was killed in September 1822 when he was captured by pirates in the West Indies, while returning from the couple's honeymoon there. She may also have been "lost at sea" in the same incident, as some accounts tell their tale. If so, she was just 20 years old at her death. She doesn't always turn up in lists of Elias Carr's descendants, perhaps because she had no children in her short life, thus no descendants.
(Note that both Winnefred and Mary married in the year their father died.)
Three Mordecai students, solid identifications on all of them. Only the fate of Mary seems even slightly uncertain.