It´d be helpful for You to know which tracking-technologies are used by that “program” - it all depends on your threat model.
And I think it is best to choose an option for protection that is practical and well enough understood (or skip the app).
You´ll find out which counter-measures (blocking, denying permissions etc.) will break the function that you want from that app. Then You can decide whether You are willing to accept that tradeoff or not.
There´s a few easy to do options that you can choose from for your local Android.
Note there´s only one VPN-interface available in your Android which means only one app can use that interface at a time, it´s either or, you have to choose.
As pointed out by @nanabanaman you can simply use Advanced Privacy’s Tracker Blocker and - utilizing the VPN-interface of Android - back that up with an app for DNS-based blocking (such as nextDNS, personalDNS, blokada or other apps alike.
That´ll be a good basic protection in my opinion.
There´s VPN-services that also offer integrated DNS-based tracker-blocking (these apps will require the VPN-interface of your Android). That will combine tracker+ads blocking and VPN. I can´t say whether a VPN-service is the tool to go with regards to tracking technology of that specific “program”.
Advanced Privacy also allows you to Fake Location and Hide my IP (the latter component will make use of the VPN-interface as well as of TOR-network which will slow down the connection), details see here
PiHole (adguard home etc.) is an option that will work only (if not configured for access via VPN) in your home-networks-environment but will also cover all other devices in your home-network. Whilst being in your home-wifi this would be more or less redundant with a DNS-blocking app on your phone.
There are surely expert-options with rooted phones or other apps or combinations of VPN+pihole etc. that others might be experienced with and willing to share…