Davenport is in Lincoln County.The Oklahoma Newspapers Archives catalog of newspapers on microfilm in the Oklahoma History Center shows this for Davenport during that time:
Newspaper title:“New Era” Roll No. 531-65:From 12/01/1910 thru 12/31/1914.It appears to be a monthly newspaper.
This newspaper seems to still be in publication, monthly.By doing a Google search for Davenport Oklahoma New Era Newspaper, several links came up which include an address and phone number.I did not find information about an e-mail or web site.
The Research Library of the Oklahoma History Center does charge a research fee.It is higher if you live out of state.It might be worthwhile to try contacting the newspaper in Davenport.