Explanation:

Correct option is 2: Lucy has already

 The structure for the past perfect tense is had + [past participle].
When you're talking about some point in the past and are referring to two events then the event that occurred first must be in the perfect tense.
The story above is written in the past tense. 'Lucy cut the cake' is the first event and
'their arrival' is the second event. So we have to use past perfect while talking about the first event.
 The other options are wrong as they make the sentence grammatically incorrect.
 Has already must be replaced with Had already. (Past perfect tense)
Hence, the correct sentence is: 'Lucy had already cut the cake when we reached the party hall.'