Can support gem effect another linked support gems?

Technically no, but practically sort of.

Supports can't affect other supports, they can only affect skills. But when supports affect a skill they change it in various ways, and that can mean the changed skill can be affected by more supports or a different set of supports.

In your example this isn't happening, because Summon Skeleton can already be affected by Minion Damage support. SPoK just changes the skill by giving it more minions it can summon, and those are naturally affected by the extra modifier to minion damage the skill gets from the Minion Damage support. Both supports are affecting the skill, not each other.

If the skill used was something like Fireball instead of Summon Skeleton, then that base skill can't be affected by minion damage normally. But the SPoKL support changes it from a skill that shoots a fireball to a skill that shoots a fireball and creates minions, and thus allows minion damage to apply to that skill, letting it modify the damage of the minions - again, both supports are applying only to the skill, but in this case, one of them applying is changing the skill in a way that allows the other to apply.

So you can in general think of the supports as applying to each other if you want, and in most cases you'll still have a good understanding of the final result, but that isn't really happening. Supports only affect skills.