A clip operation should be cutting the features that cross the clip layer at the boundary - that's what clip means. Based on a tutorial in the documentation (section 9.4.20), QGIS does support selected feature input to geoprocessing, at least in this case. Make sure your districts are on top in layer order and select the target district. Then run Vector > Geoprocessing Tools > Clip. Districts is clip layer, contam is input layer. Check the Use Selected Features box for the clip layer, so it only uses the district you want. Make sure the Selected Features box for the contam layer is not checked. Give a new file name for the output, and add it to the map when prompted. This should be contam polygons within the district, including the intersecting portions of those that cross the boundary.
If you want the entire original contam polygons that are partially within your target district, not just the parts of them within (ie, no cutting), you need to do a select and export and not a clip. Select all contam that intersect your target district and save/export them to a new file.
Should neither of these methods be producing what you want, please provide screenshot(s) showing exactly what you have, the result you're getting, and try to indicate what the result you want should look like.