It doesn't look like Frank was the father of Renie Riano. Everything else about the family story seems correct, though.
He was married in 1908 in NYC (see earlier reply), and in 1910 had no children. Renie was born in 1899 in London.
In 1910 he was a time keeper, and by 1918 he was blind in one eye.
Here are a few things I found for Frank. The 1915, 1925 census might show children to the couple, but I don't have ancestry.com. I haven't found them in 1920.
1910 Census, Bronx
914 Summit Avenue
Frank J Dobbins, 30, born in New York, parents in Ireland, time keeper for a railroad contractor
Catherine J Dobbins, 22, married for 2 years, no children
https://www.familysearch.org/ark:/61903/1:1:X39X-RYN-----------
NYC 1915 same page and house #
Frank and Katherine Dobbins
https://www.familysearch.org/ark:/61903/1:1:K92V-KYShttps://www.familysearch.org/ark:/61903/1:1:K92V-KY3ancestry.com has the images
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NYC 1925 same page and house #
Frank and Katherine Dobbins
https://www.familysearch.org/ark:/61903/1:1:KS3K-MKHhttps://www.familysearch.org/ark:/61903/1:1:KS3K-MKWancestry.com has the images
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WWI Draft Registration
Central Park Arsenal, Sep 12, 1918
Frank John Dobbins
b October 12, 1879
lives at 347 West 58th
spouse Kathrine at same residence
owns a trucking business at 246 West 46th, NYC
physical description: medium height, slender build, grey eyes, brown hair - right eye blind
https://www.familysearch.org/ark:/61903/1:1:K6J8-Q9J----------
1920 Passport Application of Catherine Dobbins,
husband Frank Dobbins was born in Fulton, NY and now resides at 346 West 48th [check my transcribing, doesn't match his 1918 address, but is close enough to be a mistake]
note: "husband's baptism certificate seen"
she's going to England to see her parents, Ireland to see friends, France to see friends - but a lot seems to be crossed out as well
sails June 10, 1920
picture of Catherine
not much more about Frank
https://www.familysearch.org/ark:/61903/1:1:QV5B-Z8XL------------
New York City Municipal Deaths, 1795-1949
Name: Frank Dobbins
Event Type: Death
Event Date: 25 Feb 1926
Event Place: Bronx, New York, New York, United States
Gender: Male
Age: 46
Father's Name: James
Mother's Name: Mary Cryan
Spouse's Name: Kate Dobbins
https://www.familysearch.org/ark:/61903/1:1:2WG4-JTR