Ah, now, if you had said that at the outset I could have saved myself the effort of looking them up. I do hope that you have actually looked at the originals rather than just looking at some unreliable online transcription of notoriously questionable accuracy, for example Ancestry. Mark's age in 1841, for example, may be a mistranscription because it is out of kilter with his age in later censuses.
That's probably why you think he was born in 1811. Ancestry and other web sites out there often (mis)'calculate' the date of birth by subtracting the age on the census from the year of the census. This gives a wrong answer three times out of four, because the census is taken about a quarter of the way through the year, so only a quarter of people have had their birthday in census year.
The available original UK census never gives a date of birth.
In the apparently almost complete absence of any baptisms or marriages of male Summers in Hamilton before 1800, finding Mark's father is likely to be a very difficult task indeed.
If Janet Barr(y) was an adherent of the Church of Scotland, and her pregnancy came to the attention of the Kirk Session, it is possible (almost probable) that the minutes of the Kirk Session might contain a record of her being summoned and interrogated about who the father was.
The Hamilton Kirk Session records for the period covering Mark Summers' birth are in the National Records of Scotland, and the volumes you would be interested in are CH2/465/11 covering 1790 to 1807, and CH2/465/12 covering 1807 to 1819.
The snag is that although they have been digitised, they are not available online. (Yet). At present you would need to go in person to the National Records of Scotland in Edinburgh, or to one of the local archives that have access to the NRS' digitised records, or hire a professional searcher* to search them on your behalf. Or you can wait in the hope that it won't be much longer now before Scotland's People add them to their resources, as they intend.
*You would be extraordinarily lucky if you were able to find a kind soul willing to do this for you free of charge. Even if you had an exact date of birth, trawling through a Kirk Session minute book from nine months before the birth to several years later is a very time-consuming task, and finding Janet Barr(y), assuming that she is mentioned at all, could involve starting at about 1804 and going on until at least the end on Vol 12 in 1819. If she isn't mentioned, your hypothetical kind soul could find her/himself spending a whole day or even more with no result to show at the end of it. That's a very big 'ask' for a volunteer.
If I were you, I would take a look at the death certificates of anyone surnamed Simmers, Sommers, Summers or Symmers in Hamilton and see if they provides any clues. From almost none in the 18th century, there are suddenly 7 S*mmers couples having children baptised after 1810 in Hamilton. There are 59 individuals in the FreeCEN transcription of the 1851 census of Hamilton, in at least 12 households. Of the heads of household, 5 were born in Hamilton, 2 in Dalserf, 4 in Cambusnethan and 1 in Ireland. But that sort of research really needs access to a Scotland's People Centre unless you are fairly flush with funds, and it may not actually yield anything useful in the end.
