Your theme determines if the player will appear on the home page or not.

If you don't see it, you can try enabling the option in SPP Settings -> Advanced Tweaks section called "Force Audio Player on Home Page"

If you are seeing random text instead of the player on the home page, you can try to enabling the option SPP Settings -> Podcast Player -> "Disable Player and Text from Home, Blog, or Archive Pages"

Otherwise you can try this free plugin called Advanced Excerpts to see if you can force the player to appear (you may need to play around with their settings to make it work with your theme):
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