"sensitive undertaking" is correct and usable in written English.
You can use it when referring to a task that requires delicacy or caution. For example, "The committee was assigned the sensitive undertaking of deciding which employees to cut from the payroll."
"It is a delicate and sensitive undertaking," Mr. Law said.
The performance was a risky and sensitive undertaking, with the band themselves frequently in tears on stage, and psychologists standing by to attend to audience members if necessary.
Reform was overdue, but it was a highly sensitive undertaking that evoked strenuous response from the services.
The authors concluded that screening a large sample to identify children and youths with psychopathic traits has some important advantages but, on the other hand, it is a sensitive undertaking because of the label 'psychopath' can have negative consequences for the subjects.
It's arguable that this is the most sensitive architectural undertaking that this country has ever seen.
Operations to gather intelligence on the Russian nuclear program are among the most sensitive missions undertaken by the United States, and disputes over exactly what is occurring on Novaya Zemlya have divided intelligence analysts and administration officials in past years.
To ensure our intervention was acceptable and culturally sensitive, we undertook 11 focus groups with a total of 117 service users, members of the community and service providers.
G4S, the company at the centre of the Olympic security fiasco, has started to recruit staff to carry out criminal investigations for a police force, with duties that include house-to-house inquiries, giving evidence in court and undertaking "sensitive high-profile cases under limited supervision".
Undertaking sensitive HIV research in the culturally and religiously diverse Pacific nation of Papua New Guinea (PNG) requires careful consideration of myriad cultural, social and religious beliefs and practices [ 1- 4].
A European diplomat who has undertaken sensitive United Nations assignments in Iran for the past two decades called Iran's push for the bomb "contradictory behavior".
It says the jobs would be "ideally suited to experienced ex-police" and successful applicants would be required to "undertake sensitive high-profile case inquiries, be able to carry out any role within a major incident room, and have specialist knowledge on all aspects of crime investigation".
