Which one sounds more natural:
Our club is having an event this Sunday.
Our club is holding an event this Sunday.
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I took a look at the definitions of both verbs to see if I could ascertain a difference but in this context, they seem pretty much the same:
Perform the action indicated by the noun specified (used especially in spoken English as an alternative to a more specific verb)
‘he had a look round’
_‘the colour green has a restful effect’
4.1 Organize and bring about.
‘are you going to have a party?’
[with object] Arrange and take part in (a meeting or conversation)
‘a meeting was held at the church’
(From the Oxford Living Dictionaries)
I wish I could give a more definitive answer, but the best I can do is give my interpretation as a native (American) speaker.
If I read that your club is having an event, I would assume that it was primarily for the members and their guests. Whether or not members were staffing the event, the intended audience would be the club's members. Examples might be a holiday party or picnic, a planning meeting, or an educational forum.
If I read that your club is holding an event, I would assume that the club members would be organizing and staffing the event, but that the target audience would be non-members. Examples would include a community outreach event to let the public know about the club and its activities, a fundraising event to raise money from non-members, or an educational event to share the club's expertise with the public.
So, to sum up:
Hold = external focus
Have = internal focus