There could be a few different things going on. Do you know the ages of the chickens? Is perhaps the white one a little young? Mature chickens will sometimes pick on immature chickens a lot. Dominance in chickens is determined more by the spirit and personality of the chicken than by size, but immature chickens don’t have the maturity to hold their own dominance wise in the flock. I’ve had chickens as young as 15 weeks be mature enough to stand up for themselves, but that is really rare. I’ve had some almost twice that old before they matured enough to stop the bullying and find their place in the pecking order. It just depends on the individual chickens involved.
Another possibility is that chickens can be territorial. They learn who is a member of the flock and might defend their territory from them. You’d think the established red chickens would defend their territory from all the new black and white new chickens, but I don’t know what is going in in there birdbrains. It might be color, but I suspect personality or maturity. This is where housing them within sight of each other for a few days can help. A lot of times this does not happen, but it can.
Sometimes the flock will attack a weak chicken. They can be real bad bullies. This weakness could be an illness or injury, it could be a young chicken without the will to defend itself sort of like I talked about above, or it could just be a mild-mannered chicken. A lot of times, there is one specific hen that leads the attack. Once she starts, the others join in.
Possible remedies, depending on what is actually going on:
Understand that a pecking order has to be established so they know their place in the social structure. If no blood is drawn, they can probably work it out. But if blood is drawn, you need to intervene.
If you can house then side by side for a few days, where they can see each other but can’t get to each other.
Give them as much space as you can. Thee way the pecking order usually works, if a chicken invades the personal space of a more dominant bird, the dominant one has the right to peck the inferior to put it in its place. The inferior runs off and the proper order has been restored. If the inferior does not have enough space to run away, that is considered a challenge to the dominant one and a serious fight can happen. Also, if space is tight, they will invade that personal space more often. This is why you often see separate flocks when you try to integrate new chickens. They are trying to stay out of each other’s way.
Give them extra places to eat and drink. This cuts down on conflict points.
If you can identify a ringleader in these attacks, isolate that ringleader for several days. This can knock her down in the pecking order so when she comes back she has to regain her status in the pecking order. This way, she does not pick on one specific chicken but has to look out for the rest of the flock. Some people on this forum say that their dominant hen is the one doing this, but when I’ve observed this behavior, it’s been the hen lowest in the established pecking order. I’ve never had this type of problem with a mature rooster. It has always been a hen or immature cockerel.
Another possible remedy, if you can manage the space, is to only add one or two of the more docile hens to the newcomers at a time. Let them work it out before you add more aggressive chickens. This way the ones being added face a more united flock. And if possible, add them on the new chicken’s home turf so the others don’t feel like defending their turf.
I don’t know if any of these will help. Each chicken has its own personality and we all have different set-ups. I do think space is real important. If they don’t have room to get away or avoid each other to start with, integration is a lot harder. You can maybe create some space by giving them places to hide behind or under. Often, I’ll see the immature ones staying on the roost in the coop while the mature ones are on the ground. They are just trying to stay out of the older one’s way.
I wish you luck. Sometimes integration goes so smoothly you wonder what all the worry was about, and sometimes it gets really bad.
20 weeks while the reds are at about 17-18 but yes the red one mounts the poor thing too 
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