"I have a white pony."
Translation:Minulla on valkoinen poni.
Are you asking why on is placed second in the sentence, or why on is in the sentence at all?
On needs to be included because it's the only verb in the sentence. Minulla isn't a verb. Literally, minulla just means 'on/at/by me'. Finnish doesn't have a verb 'have', so it uses the construction 'At me is a white pony'.
As for word order, I believe the most neutral order in statements puts on between the subject and whatever noun, adjective, or adverb on is linking the subject to.
So both 'Minulla on valkoinen poni' and "Valkoinen poni on minulla' are possible, though the latter is more likely to translate to "I have the white pony".