This agreement was damaging to Britain because it hurt Britain's relations with Arab countries and, perhaps more importantly at the time, its relations with the United States.
Before this time, Britain had decent relations with Arab nations. The Sevres Protocol, however, showed Britain colluding with Israel to oppose the aims of an Arab country. This harmed British prestige in the Middle East and left it without any real political influence in the region.
At the same time, this hurt relations with the US. The US was opposed to colonialism (although it had done things like supporting the French war in Vietnam) and did not like what Britain had done. It also did not like the fact that the British actions helped make US relations with the Arab countries more difficult. The US wanted good relations with the Arabs to prevent Soviet influence from growing in the region.
In these ways, knowledge of the Sevres Protocol hurt Britain by damaging its relations with various foreign countries.