A request from Ikora part III, already done weekly heroic, will it count if I do it again?

I've done the Weekly Heroic strike already this week.

Just now I completed "A request from Ikora Part II" and got Part III, which says "Complete a Weekly Challenge Activity".

Can I do the weekly heroic strike again this week and get that bounty completed (or moved on to the next step), or does it only count for this bounty if it is the first time I do it this week?

Yes it will count for Ikora part III, since you are still doing the weekly heroic strike, there is no limitation if you have already done it this week.

Just remember you won't get any loot for it, it will only count for the mission.

Source