Figures show the Royal Devon University Healthcare Foundation Trust is not meeting a key cancer target.
A think tank has welcomed "green shoots" for cancer care, but warned the NHS still has work on its hands to reduce the overall waiting list.
Figures show cancer patients at the Royal Devon University Healthcare Foundation Trust are not being seen quickly enough.
The NHS states 85% of cancer patients urgently referred by a GP should start treatment within 62 days.
But NHS England data shows just 67% of patients urgently referred by the NHS who received cancer treatment at the Royal Devon University Healthcare Foundation Trust in March began treatment within two months of their referral.
That was up from both 65% in February, and 61% in March 2023 last year.
Despite this, across the country a different cancer target was met for the second month in a row.
Some 77.3% of patients in England urgently referred for suspected cancer in March were diagnosed or had cancer ruled out within 28 days.
This is down from 78.1% the previous month, but is the second time in a row the target of 75% has been exceeded.
The King's Fund think tank said there were "green shoots" for cancer care across England.
However, chief analyst Siva Anandaciva cautioned: "the road ahead to make further progress in recovering performance in other areas of the NHS, including reducing lengthy waiting times for planned care and A&E, will be long".
Separate figures show 73,539 patients were waiting for non-urgent elective operations or treatment at the Royal Devon University Healthcare Foundation Trust at the end of March – down slightly from 73,737 in February, and 82,007 in March 2023.
Of those, 3,798 (5%) had been waiting for longer than a year.
The median waiting time from referral at an NHS Trust to treatment at the Royal Devon University Healthcare Foundation Trust was 16 weeks at the end of March – the same as in February.
The Nuffield Trust health charity also welcomed progress on cancer care, but warned progress on cutting the overall waiting list had "stagnated".
Acting director of research Sarah Scobie said: "While the NHS has made some progress getting those who have waited longest treated, it hasn’t been able to bring overall numbers down.
"The high numbers of patients waiting for care and the progress to get people seen faster will remain under intense scrutiny leading up to a general election."
Politicians should avoid "unrealistic and bold" commitments on waiting lists, Ms Scobie added.
Nationally, 7.5 million people were waiting to start treatment at the end of March.
In addition, 1.6 million patients in England were waiting for a key diagnostic test in March – the same as in February.
At the Royal Devon University Healthcare Foundation Trust, 16,739 patients were waiting for one of 14 standard tests, such as an MRI scan, non-obstetric ultrasound or gastroscopy at this time.
Of them, 5,287 (32%) had been waiting for at least six weeks.
Health and Social Care Secretary Victoria Atkins said: "NHS staff are working tirelessly to cut the waiting list and today’s data shows the biggest six-month reduction in over 10 years outside of the pandemic.
"This is a significant achievement in the context of record pressures and strikes, with NHS analysis showing the list could have fallen by an extra 430,000 since December 2022 without industrial action.
"We’ve also delivered on our target of ensuring over 75% of patients tested for cancer receive a diagnosis or all clear within 28 days of referral – giving patients the all-clear or a diagnosis sooner."