Ms Gibson said she had been approached, while using health services as a patient, by staff who wanted to raise issues with her in confidence.
Several members of staff also attended Wednesday’s debate, which was called after a 24-hour strike by HCRG workers over staffing levels, workloads, and concern for patient care on October 21.
Bristol South Labour MP and Minister for Secondary Care, Karin Smyth, said: “Single points of access have been worked on for many years under many different contracts. Generally, they should do what they say on the tin and actually help people move forward, so HCRG does need to make its work.
“I understand that, as a result of some of the issues that have been raised, the ICB (integrated care board) is increasing its oversight, bringing together the provider and the system to support improvement. It is really important that that work also brings together patients, carers, the local community, staff and other stakeholders as it develops.”