What was the significance of the Suez Canal crisis to the relationship Britain had with its Empire?
The significance of this crisis can be argued either way.
On the one hand, there is clearly evidence that the crisis eroded Britain's ability to control the empire. In the Suez Crisis, Britain looked exceedingly weak. This was noticed by countries that could be seen as part of Britain's "empire" at that point. In particular, countries of the Commonwealth, such as New Zealand and Australia, did not support Britain diplomatically during the crisis. India publicly sided with Egypt.
On the other hand, it is not as if Britain lost all influence in her colonies and spheres of influence because of the Suez Crisis. Britain still had troops in the Middle East and continued to exert influence through membership in the Baghdad Pact. Britain did not lose any African colonies as a result of the crisis.
Therefore, the evidence is mixed. The Suez crisis appears to have had some effect on Britain's relationship with its empire, but it did not devastate that relationship.