An Esrog with a hole that has a missing piece:
A. First Day of Sukkos:
A very small amount is missing: An Esrog which is missing any part from its body, even the smallest amount, is Pasul. This applies even if the hole does not reach from one to the other and is not the size of an Issur. However there are opinions which rule that even a hole with a missing piece is valid if it is not larger than an Issur and does not go from side to side. Practically one may rely on this opinion if there is no other Esrog available and one may use it with a blessing.
In a case of doubt: If one is in doubt whether a piece is missing from the Esrog or if it is just an indentation the Esrog is permitted [so long as the hole is not the size of an Issur and is not Mefulash].
A large amount is missing: If the missing piece is as large as the circumference of an Issur coin it is Pasul according to all opinions.
A piece is missing from the outer membrane:
The above is only with regards to if there is a piece missing from the actual body [white thick peel] of the Esrog. If however the Esrog is only missing part of its thin outer membrane [i.e. skin/rind] and its inner white thick peel remains complete, the Esrog is kosher. If however the entire outer peel is missing the Esrog is invalid.
If the missing membrane causes color change: If the missing outer membrane has caused a change in color which differs from the color of the rest of the Esrog it receives the same Halachic status as a blister, and hence the Esrog is invalid unless the color change is not within the Chotem of the Esrog and is not majority of the Esrog and it is not within 2-3 areas of the Esrog.
A hole which was caused by a thorn while the Esrog was still on the tree is kosher even if a piece is missing, if the entire area of the hole is covered by scar tissue. This applies even if there is still a piece visibly missing from the Esrog being that the missing area is fully covered by the scab. If however the scab is not fully covering over the hole, or there is no scab at all, and one can tell that there is a piece missing, the Esrog is invalid.
B. Yom Tov Sheiyni Shel Galuyos:
On the second day of Sukkos in the Diaspora if an Esrog is missing a piece the size of an Issur coin one may use it if no other esrog is available, but without a Bracha.
C. On Chol Hamoed:
On Chol Hamoed an Esrog is Kosher even if it is missing a piece, and even if the missing piece is much larger than an Issur coin so long as the Esrog retains its minimum size.
If an Esrog is missing a piece, irrelevant of its size, it is invalid on the first day of Sukkos unless either:
- The piece is only missing from its skin and not its body, and this missing area is not a different color than the Esrog.
- The piece was removed while the Esrog was on the tree and a scar grew over it.
 Source of these invalidations: All of these invalidations of a hole, and crack is not due to Hiddur but rather because the Esrog is not considered Tam [complete] and the Torah states “Ulikachtem” and the Sages expounded this to mean a full and complete taking. [648/14]
 See below for the difference in whether it is missing from its body or merely its skin.
 The reason: In such a case of doubt one may be lenient even initially to use the Esrog as there are opinions which rule that even if a piece is missing it is valid so long as it is not Mefulash and is not an Issur. [ibid]
 Other Opinions: The Biur Halacha rules that only if the wax layer covering the green is missing it is valid. If however the inner green thin peel is missing it is invalid. [Piskeiy Teshuvos 648/4]
The reason: The reason for this is because once the Esrog no longer has its external peel it becomes ugly and is no longer Hadar. Some rule it is only Kosher if it retains an Sela size of peel. Others rule even if the most minute amount remains, it is Kosher as due to that small amount the Esrog will return to its healthy status. Practically one is to suspect for the first opinion when possible. [ibid]
 The reason: As since the hole occurred while on the tree and the scab grew over the hole and covered it, it is considered like a normal growth over the Esrog and hence it is Kosher. [ibid]