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Galaxies that reside in galaxy clusters are know to have peculiar velocities that cause the Fingers of God effect.
But what about galaxy clusters themselves?
Do they also have peculiar velocites by interacting with other galaxy clusters?
If so, are they high enough to be taken into account or low enough to be ignored?
I'm going to break your question down into two sections.
Do galaxy clusters have measurable peculiar velocities?
The answer to this depends slightly on what is meant by "peculiar velocity". In the most general sense, the peculiar velocity of some object is the velocity it has with respect to some standard of rest. In the cosmological sense, the peculiar velocity of a galaxy in a cluster is simply any motion that galaxy may have which is not attributable to the Hubble flow. Primarily this peculiar velocity is a result of that galaxy's orbit around the cluster barycenter.
With that in mind, it should hopefully be easy to reason out that clusters themselves will have peculiar velocities. It is highly unlikely that a cluster will be "fixed" in space and only move according to the Hubble flow. A quick google search revealed Bahcall & Oh 1996 which cites an rms one-dimensional galaxy cluster peculiar velocity of $298\pm28\:\mathrm{km/s}$ (based on 22 clusters). While this is smaller than a typical galaxy's peculiar velocity, it is certainly measurable and capable of being distinguished from the Hubble flow (at least for reasonable redshifts).
The real difficulty here is not whether the cluster has a non-negligible peculiar velocity, but rather, is that peculiar velocity even measurable due to many of the potential limitations. While the value of the peculiar velocity of a cluster is certainly "high enough to be taken into account" it can often be difficult to measure such a velocity with any reasonable error (a point that Bahcill and Oh discuss).
Can these galaxy cluster peculiar velocities produce the "Fingers of God" effect?
To answer this you have to consider a very particular point I made in the last section which was that for galaxies the primary contribution to their peculiar velocity is their orbital motion around the cluster barycenter which can result in peculiar velocities on the order of $10^3\:\mathrm{km/s}$.
When trying to map out a particular cluster in redshift space, you can often get "Fingers of God" in your maps which are a result of the cluster appearing elongated towards the observer and due entirely to the orbital peculiar motion. An example of such an effect is below, with the red circles added by me to indicate these fingers of god.
Galaxy clusters on the other hand are not gravitationally bound to other galaxy clusters! This means any peculiar motion a cluster has is simply its general motion through space. Since the fingers of god are a result of orbital peculiar motion, it stands to reason that galaxy clusters will not have such a motion and as such clusters won't produce these fingers of god. You'll find that more or less, the peculiar motion of clusters will simply move cause them to be randomly re-positioned without any real "structure" that you see in the fingers of god for galaxies.
I think the key point is how the velocity dispersion in a given structure compares with the velocity dispersion you would expect from the difference in the Hubble flow across the linear extent of the object. Or, equivalently, what range of distances (according to Hubble's law) are implied by the velocity dispersions and how does this compare with the actual size of the structure in question.
Fingers of God effects in clusters are produced because the velocity dispersions ($\sim \pm 1000$ km/s) would imply a front-to-back distance of $\pm 14$ Mpc (for $H_0 = 70$ km s$^{-1}$/Mpc), compared with the actual size of a cluster that typically only has a diameter of $2$ Mpc.
Galaxy clusters may themselves be part of superclusters. These superclusters may have sizes of order 100 Mpc or more. However, the peculiar velocities of clusters at similar distances (that can be judged from the average recession velocities of their members) is actually smaller than the peculiar velocities of individual galaxies. Thus the change in Hubble flow velocity across a supercluster is much larger than the peculiar velocities of the individual members (andthey are not bound together). So I would not expect a Finger of God effect in the members of a supercluster.
