They are usually called clinics not centres in British hospitals, but that will depend on context.

Dedicated to the treatment of terminal stages of certain illnesses like terminal cancer.

This translates literally as "pain center" and would refer to a center or clinic that dealt with pain management or study.

Se si riferisce alle terapie relative ai tumori va' ancora meglio Cancer Pain Management Center.

A centre where Pain is managed, especially in the oncology sector

Both "pain management center" or "pain management clinic" may be used in American English. For instance, St. Elizabeth's Medical Center in Boston uses "Pain Management Center". Methodist Medical Center in Dallas refers to the center at their facility as the "Pain Management Clinic". Neither center/ clinic places any special emphasis on oncology, but rather, all pain syndromes are treated equally. For example, St Elizabeth's Pain Management Center writes: "Our comprehensive pain management program offers evaluation & treatment of a wide range of pain complaints including back pain, sciatica, headaches, diabetes-related pain, fibromyositis, stress-related pain, chronic fatigue syndrome, polymyalgia, muscle pain,disc problems, pain from damage to nerves, shingles, HIV/AIDSrelated pain, cancer-related pain, TMJ, arthritis pain, post-operative pain, and post tramatic stress disorder." The NYU Pain Management center and the Methodist Medical Center in Dallas describe the goals of their respective centers/clinics in a very similar manner, as does the "Centro del dolore dell'Università di Milano"; for example the Milan facility describes its center as, "una struttura ospedaliera progettata in modo specifico per lo studio e la terapia dei pazienti con dolore cronico, tumorale e non tumorale...comprende 7 ambulatori...una sala per brevi degenze... polo operatorio completo... una sala sterilie per blocchi antalgici...infine C'è un day-hospital con 3 posti letto." I hope this helps.