There are at least a few approaches:
1. Don't worry about it. Seriously, if you're just doing clustering and looking at marker genes between clusters in the same sample, then a bit of background contamination isn't a big deal. Any contaminating gene probably won't be DE if the contamination is more-or-less even across cell types (assuming similar total RNA content in each cluster), so it won't show up in your marker lists for clusters that aren't actually overexpressing it.
2. Identify and remove the affected genes from your results, e.g., DE tables. This may be necessary when dealing with multi-sample comparisons where the ambient contamination might differ between conditions. In such cases, it is possible for the ambient contamination to drive false DE between conditions, so getting rid of them is important.
3. Try to remove the ambient contamination from the expression matrix at the start of your analysis. This can be done by packages like SoupX, but I would say that this is the most challenging approach. IIRC, you need to have prior knowledge of at least two markers that are highly abundant in the ambient solution but can never be expressed together. This is used to identify the proportion of ambient contamination in each cell, which is unlikely to be very precise given the level of per-cell noise. Then you have to perform the actual process of subtracting counts, which can be difficult due to the mean-variance relationship of count data.
You could use clustering or nearest-neighbors information to overcome some of the stability and variance problems in 3, but if you're already got clusters that suitably summarize your data, it seems pointless to go back to edit the expression matrix. (Well, aside from improving the aesthetics of your plots.) The bigger problem is that I never have two genes that I am absolutely sure are never co-expressed. Even identifying one gene that should not be expressed in a single subpopulation of the data is hard enough... unless you're working with very well-established cell types, but if so, you shouldn't need to bother removing contamination at all.