If you send me your e-mail address in a PM, I can send the passport application to you.
When Rebecca sailed to the US in 1914, she was with children Elizabeth, age 17; John, age 10; and Mary, age 8.
The daughters are with them in Rochester on the 1920 Census. They were still living at 124 Comfort St.
1920 Census
Paterson, Alexander, head, rents home, male, white, age 43, married, immigrated 1914, papers filed for naturalization, able to read & write, b. Scotland, parents b. Scotland, occupation - dyer
wife Rebecca, female, white, age 43, married, immigrated 1914, alien, able to read & write, b. Scotland, parents b. Scotland, no occupation
daughter Elizabeth, female, white, age 22, single, immigrated 1914, alien, not attending school, able to read & write, b. Scotland, parents b. Scotland, occupation - clerk, dyer work
son John, male, white, age 15, single, immigrated 1914, alien, attending school, able to read & write, b. Scotland, parents b. Scotland, no occupation
daughter Mary, white, age 13, single, immigrated 1914, alien, attending school, able to read & write, b. Scotland, parents b. Scotland, no occupation
If you can get exact birth dates for the children, it might help to find them.