Praise be to Allah.
Firstly:
With regard to the person who is going to offer an udhiyah removing anything from his hair, nails or skin, this is an issue concerning which the scholars differed. The more correct view, as stated on this website, is that it is not permissible to remove anything from the hair, nails or skin for the one who wants to offer an udhiyah, because the Prophet (blessings and peace of Allah be upon him) said: “When you see the new moon of Dhu’l-Hijjah, and one of you wants to offer a sacrifice, let him leave his hair and nails alone.” Narrated by Muslim (1977). According to another version also narrated by Muslim: “When the ten (days of Dhu’l-Hijjah) begin, and one of you wants to offer a sacrifice, let nothing touch his hair or skin.”
For more information, please see the answers to questions no. 83381 and 36567.
Secondly:
If refraining from removing hair from the moustache will look odd or cause embarrassment – as you mentioned – because of there being a spot where no hair grows, then what appears to be the case – and Allah knows best – is that it is permissible for you in this case to remove hair from the moustache, so as to avoid embarrassment.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: If a person needs to remove anything from his hair, nails or skin, then there is no blame on him for removing it, such as if he has a wound and he needs to trim the hair from around it, or if he breaks an nail and it bothers him, so he trims whatever is bothering him, or if he has a piece of dead skin and it is bothering him, so he cuts it off. There is no blame on him in any of these cases.
End quote from Majmoo‘ Fataawa Ibn ‘Uthaymeen (25/161)
And Allah knows best.
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