The United Lincolnshire Hospitals Trust has asked residents to complete an online survey to "understand the public's perception and how people form opinions about ULHT hospitals … with the goal of making improvements".
The Care Quality Commission recommended maintaining confidence in special measures over the past year, with executives telling inspectors they were "consumed by operating pressure."
However, the inspectors stated that the Trust had made "great progress" and increased its overall rating from "insufficient" to "in need of improvement".
The Foundation consists of Lincoln County Hospital, Pilgrim Hospital, Boston, Grantham and the District Hospital and County Hospital in Louth.
During a CQC visit to the Pilgrim Hospital Emergency Department, which was conducted after concerns over the standards had been voiced, children in the ward were found to be at risk of injury.
HSJ Also last year, it was reported that Unite's staff committee at the Trust had issued a vote of no confidence due to patient safety concerns.
ULHT CEO Jan Sobieraj, who is retiring this summer, said the Trust wants to find out what the public thinks and how "trustworthy and credible we are when we spend money wisely and when people think we are quality provide high quality care ".
He added: "It is important for us to measure our reputation as it interferes with our ability to respond to the diverse needs of our patients, their caregivers and families, and help us recruiting the best people.
"As we build on the quality of our services, our CQC rating has improved, and our reputation for care and safe patient care is growing as a result. But we can do much more to make it better. "