Marina Corrigan told the tabloid she believed she was asked to leave her post as area manager for Kent and Sussex because she would be absent from work while caring for her 16-year-old son Ashley.
Corrigan's son was diagnosed with advanced testicular cancer on 20 April.
A month after the diagnosis, and five days after Ashley had begun chemotherapy, Corrigan lost her job.
Corrigan, a single mother, said: "When we found out Ashley had cancer, I was heartbroken. But I thought my employer would understand better than anybody exactly what I was going through. How wrong I was."
The trust, an official charity of next year's London Marathon, said that as a result of the article, it received a call from a supporter who asked to be taken off its register. However, Dr Steve McEwan, the charity's chief executive, called the supporter and persuaded them to stay on board.
A spokeswoman for the charity said: "We save lives, and this bad publicity is costing people's lives. Prior to her son's illness, Marina Corrigan had her initial probation period extended for performance-related issues.
"At the end of that, the decision was made not to confirm her in the post because they had not been addressed."
As Corrigan lost her job before completing a year's service, it is unlikely that she would have a case in an employment court, according to Simon Kerr-Davies of Cripps Harries Hall LLP.
He said: "Although you might expect compassion from an employer in this situation, you would have no formal protection in law."