Where did Edward Harkus write from?
There are three Edward H*rk*s births between 1860 and 1870 in FreeBMD
Edward Harkis, Morpeth, 1861, 3rd quarter
Edward Harkus, Newcastle-upon-Tyne, 1862, 3rd quarter
Edward Harkis, Morpeth, 1867, 4th quarter
and a death of Edward Harkus, Newcastle-upon-Tyne, 1862, 4th quarter (ages are not included in the index in 1862 so cannot say for sure that it was the second child above). Although there isn't another death, there is never more than one Edward Harkus/Herkis/Herkess etc of this age group in the census index from 1871 onwards, and some years none at all.
The 1881 census CD-ROM transcription lists just one Edward Harkus in this age group - a 13-year-old coal token taker living in Bedlington, Northumberland, with parents James and Ann Harkus, four siblings and a 15-year-old step-brother named Peter Hepple. James Harkes married Ann Heppell in Newcastle-upon-Tyne in 1866, 4th quarter.
Edward Harkus married Elizabeth Jane Appleby in Morpeth in 1889 and they had a daughter Annie in 1890 and twins May and Christopher William in 1893.
From the 1881 census it looks as if Elizabeth Jane is the daughter of Thomas and Margaret Appleby, born in Morpeth in 1879,3rd quarter.
So is this the right Edward Harkus? Are there any clues in the letter that match any of the foregoing?
If he is the right Edward Harkus, it could mean either that Mary Duncan's father was James Harkus or that Mary Duncan's father was Thomas Appleby. Or there might be another possible relationship which has not occurred to me

However, it might be John Edward Harkus. He was born in Newcastle-upon-Tyne in 1863, 4th quarter, and the 1881 CD-ROM lists him as the 17-year-old son of Robert and Mary Harcus. Mary was aged 40 and born in Scotland. There is a marriage of Robert Harkus in Newcastle-upon-Tyne in 1859, bride either Mary Hannah Pool or Jane Thompson, and another in 1868, bride
Mary Duncan or Martha Kay. John Edward Harkus married in Newcastle in 1883, bride either Elizabeth Collins or Isabella McDonald.
This looks promising, but raises several questions. If John Edward Harkus is Mary Duncan or Gillespie's brother, how come (a) his birth was registered as Harkus and not as Duncan, when his parents were apparently not married until five years later and (b) why does he not appear as Edward in the census, if that was the name he habitually used? (I can't actually find him at all in 1891 or 1901, but that could be because of poor indexing.) Maybe he was in fact her stepbrother, a son of Robert Harkus by a previous marriage?
Food for thought?