I assume that by CWC, you refer to cluster-mean-centering of hierarchical/clustered data.
When multilevel regression models are used to analyze hierarchical data, you'd typically want to center your level 1 predictors. How to center depends on your substantial research question. IMO this is a good article about the topic of why and how to center:
https://psycnet.apa.org/record/2007-07830-001 (there's a free version in the OSF repository)
Cluster-mean centering means centering predictor values around each cluster's mean, in the case of people (participants) being the clusters, this means centering around each participant's own mean. In practice, you have a level 1 predictor x, for which you have several values for each participant. You compute the mean x separately for each participant, and then you subtract the mean from each individual x value. This results in a variable that has a mean of zero for each participant, positive values mean "higher than this person's average" and negative values mean "lower than this person's average", and all between-person differences have been removed.
You'd use participant-mean-centering if you are interested in the within-person effect for this regression part in your model. As an example, if x was momentary sociability, and your outcome was momentary mood, you'd participant-mean-center sociability if you were interested in whether participants are, on average, on better mood during the moments during which they are more sociable.
By contrast, if you are interested in the between-person effect, i.e. whether those participants who are more sociable on average are also on better mood on average (across all measurement occasions), you would use grand mean centering, i.e. centering sociability around the grand mean of sociability (around mean sociability of the whole sample).
You usually would want to use either CWC or grand-mean centering instead of just using the raw values, because with raw predictor values in a multilevel model, your estimates will be a mix of within- and between-person effects, and you can't disentagle those.
Cluster-mean centering is easy to do in R using dplyr's group_by and mutate, or using the package misty.
In MPlus you can do the centering within the model code using the CENTER command and subcommands GROUPMEAN or GRANDMEAN but I'm not sure about the exact syntax.
