Need help with flow for email campaign

I think the reason your subsequent touches are going out on Friday is because you've added the "wait must end on" logic and you are forgetting to account for processing time and/or time of entry.

Do you have this setup as a batch campaign or a trigger campaign? 

Let's assume that this is a batch campaign.

If you schedule the campaign to run Thursday's at 10:00 am that means people are going to enter your flow for the first time on Thursday at 10:00 am and then the first flow step would process (I'm assuming this is send email 1). That flow step may not finish processing everyone at 10:00 am exactly. It might take a couple minutes to actually send out the emails. (You can check the time stamps on the results tab of your smart campaign to confirm.) Let's assume that the email actually sends to a person at 10:01. Then your wait starts for that person. Under your current setup the one week wait is going to end for that person at 10:01 am the following Thursday. That means it has already missed 10:00 am so the wait continues until the following day at 10:00 am. Now if you were to remove the "wait until" requirement and just wait 1 week that means each person would move to flow step 3 (send email 2) at 10:01 am on Thursday. 

The scenario would change a bit if this is a trigger campaign because you will have people entering at different points in time.

With a trigger, you have two potential setups:

Option 1: You want the person to receive their first email Thursday at 10:00 am and then all emails after that Thursday at 10:00 am. You would use the same setup as the batch campaign above but would add a wait as the first flow step and make it very short (1 minute) and wait must end Thursday at 10:00 am.

Option 2: You want the person to receive their first email immediately and then all emails after that Thursday at 10:00 am. Here, send email would be the first flow step followed by a short wait (say 24 hours) and wait must end Thursday at 10:00 am. After the second send email you would use wait step of 1 week.

In this setup, let's assume Person A enters the flow on Tuesday at 9:00 am. They would receive email 1 at approx. 9:00 Tuesday, then email 2 Thursday at approx. 10:00 am and then all subsequent emails on Thursdays. 

If Person B enters the flow on Thursday at 9:00 am they would get email Thursday at 9:00 am and then wait until the following Thursday at 10:00 am to get email 2.