What is UECapabilityEnquiry in LTE?

It is the downlink LTE RRC message the eNodeB sends to request UE capability information.

What comes after UECapabilityEnquiry?

The UE normally responds with UECapabilityInformation.

Does UECapabilityEnquiry contain the UE capability itself?

No. It only requests the capability information. The UE reports the actual capability in UECapabilityInformation.

Why is UECapabilityEnquiry important?

It helps the network avoid applying feature-dependent LTE configuration that the UE does not support.