Does casting a spell for free through Suspend still allow you to pay kicker costs?

When a card's last time counter is removed via Suspend, its owner casts it without paying its mana cost (CR 702.61d). 'Without paying its mana cost' is a cost-reduction effect that applies specifically to the card's printed mana cost — it does not prevent you from paying other costs you choose to pay.

Kicker is an optional additional cost defined under CR 702.33. When you cast a spell, you first choose all applicable modes and optional additional costs during the announcement step (CR 601.2b). Since Kicker is a separate, voluntary cost on top of the mana cost, you are free to pay it even when the base mana cost is waived.

This is confirmed by CR 601.2b, which states that the player may choose to pay optional additional costs at announcement, and by CR 702.61d, which only waives the mana cost, leaving all other costs intact and choosable. The same logic applies to other optional additional costs like Buyback or Entwine.

Example: You have a suspended Kor Sanctifiers (Kicker {W}: destroy target artifact or enchantment). Its last time counter is removed, you cast it without paying its {2}{W} mana cost, but you may still pay the Kicker cost of {W} to get the destroy effect — you just need to produce that {W} mana normally.