Welcome there. I made a build for an AD and AP poppy. im just testing builds : but that one was the most suseccful to me. and i hope to you too.

These are the Pros and Cons that my build give :
Pros:
-Descent damage mid game.
-Both AD + AP + Crit damage = every thing you do deals nice dmg when you got feeded. :p
-Paragon of demacia + boots = chasing/running speed.
-Nice armor/magic resist (depending on what defensive item you pick.) at mid game
Cons:
-SUCKISH MANA AND MANA REGEN EARLY AND MID game.
-Maybe a little squishy on Early game.
-Bad Atack speed for turrets, or whatherver you need to. :p
-with the HP you will end, you are maybe the first/second targeted on teamfights

I start wirth some boots, with an HP potion. you can chose an MP one, if you want.
i try to build AD and a good lifesteal trying to build onto a Bilgewater Cutlass, and then for a cloak. the Hextech gunblade can wait. i go for a B.F. sword for extra AD, and converting it onto an Infinity edge. i go for a hextech revolver, then converting it onto a Hextech gunblade. I go for sorcerer shoes. i try to go for an Eleisa mircacle for HP and mana regen with tenacity. You can now chose onto :
-Hexedrinker
-Thornmail
-Zhonya hourglass
one of them gives armor, or magic armor. you can pick; by the way : all of the 3 are good options.
Go for other B.F. sword, and turning it into a Blood thrister.

Good:
-Flash: when your paragon of demacia is on CD, or when you cannt scape when you only have some normal boots, this will be a good chasing/scaping summoner spell.
-Cleanse: Nearly same use of flash, by the way its more usable if you are stunned, slowed, feared, etc. and it has a short cooldown!
-Ghost: very nice chasing spell, or simply to come onto a teamfight faster or just for godlike spped when you have your Magic penetration boots + paragon of demacia on.
Maybe:
-Teleport: Great source for poppy, that is maybe descent to teleport , because poppy needs mana in general. by the way, there are mostly useful spells.
-Surge:Gives hability power and atack speed, but its not worth it, neither for a chasing
-Fortify: Great option for poppy, even she isnt a tank or a support, but, you need some more adjustable to what poppy needs.
Bad:
-Clarity/Heal: as you knew why are they for: they just are useful for early game. no more.
-Ignite: Good for damage over time for ending a chase and killing the foe who´s scaping, but, cmon, poppy is a chaser and she dont needs that.
-Clairvoyance: Poppy its not a support or she dont has a skillshot with large range.
-Smite: i will use that in poppy to solo baron with evelyn, HUEHUEHUEHUE.
-Revive: Just for trolls and feeders, i think you understood what i said.

Hope you liked it. you can tell me constructive critics and kills/deaths/assists you got in the whole game with my build. :)