I may have figured this mess out. In W-10 (it should be similar steps, more or less, in 8 - 8.1), find the short cut in question, in the Start Menu, right click it, choose More, then choose-click on: Open File Location (in other Windows versions you might
have to use "Properties" for this... and "improvise" a little bit with some steps... ?). Now Right click the (new) Microsoft Support short cut (or what ever the name of the one in your start menu is called) and look for the: "Target" - line/box (Find Target?).
Using the details in there, you'll find the file path to a folder (a LogMeIn type folder, I think) containing a related file with a .bat file extension inside. I deleted that whole folder (with that bat extension file inside) and the earlier downloaded Support-LogMeInRescue.exe
file from my downloads folder. I guess the shortcut on the Start Menu needs to be deleted as well... This is the best I could come up with and in theory, it should block that which we all Fear!