I experience the same thing for sure. The delay varies. Sometimes it’s a second or two and others it’s like 10. Would be glad to know if anyone has an idea as to why this occurs and/or how to alleviate it!
I’m seeing a similar delay on my Homepods (OG) for TTS messages. I’d also be interested in any ways to reduce this!
Also, has anyone been able to resume previously playing media when using Piper TTS with homepods? I previously used a script to grab the current volume and content_id for apple music before playing the TTS message (similar to the scripts described here), but that doesn’t seem to be working anymore. Now playing a TTS message causes the current app to change and the previous music doesn’t resume.
Could somebody please advise which ports (TCP and UDT) need to be open between HASS and HomePod Mini for them to communicate?
I have the HASS and the HomePod on the separate VLANs. If I turn off all the firewall rules between the VLANs, HASS talks to HomePod no problem and I can do TTS. With only the following ports open the communication stops and HASS integration cannot reach the HomePod. I also have AppleTV and iPhones on the same VLAN as the HomePod. HASS has no problem reaching those with only the aforementioned ports rules open.
Airplay ports:
TCP: 7000
UDP: 6002, 49152-65535
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I am trying to figure this out as well. Disable all firewall and it works. I have tried many of the ports listed in this article from Apple (but not all) and still can’t get it to work. Maybe I will try all and cut back one by one. TCP and UDP ports used by Apple software products - Apple Support
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Any luck? 
I thought I was on to something. I would get it to work if I disabled my firewalls. I then went back and added specific ports and port ranges and would get it to work for a couple tries. Or so I thought. It turns out that it is something in Home Assistant communication to HomePods that stoped working. If I disabled the firewall and tried a few times it still stopped working. I am not sure exactly what is causing it. It always works fine to my browser window, but rarely ever works to my HomePods.
I am wondering if it has something to do with running in a Docker container. The container has lots of memory, cpu, and disk available.
I hope to upgrade to a Home Assistant Green soon and I will try from that.
I’m on RPI 3b+. Works with firewalls off, drops after some time when I turn firewalls on.
Planning to transfer to VM. Then it should be easier to trace the comm problems with Wireshark.