I have tried with various recipes to make cream cheese icing, and each time, the icing has turned out too runny to 'frost' a cupake or carrot cake. I've tried using full fat cream cheese and the correct proportion of cream cheese to icing sugar, but still no success (e.g. 125g cream cheese, 250g icing sugar, 2 tsp lemon juice). I've tried using an electric mixer and by hand. Many thanks.
Unfortunately occasionally cream cheese frosting does liquefy and this mainly comes from the liquid in the cream cheese mixing with the icing (confectioners') sugar and dissolving it, leading to a runny icing. Adding extra sugar will not correct the problem and will also lead to a frosting that is too sweet. You are correct to use full fat cream cheese as the reduced and low fat types contain more liquid (plus stabilizers) and will not work properly for a frosting. We would suggest reducing/omitting the lemon juice as this adds extra liquid - you could use finely grated lemon zest instead if you want to add a lemon flavour or a couple of drops of cider or white wine vinegar if you want some acid to cut through the sweetness of the frosting. Always add any extra liquid at the end.
There is a chance that the cream cheese was over beaten before the sugar was added, which can sometimes lead to the cheese partially melting and dissolving the sugar. You could try making the frosting in a food processor, putting the sugar and cheese in together and whizzing for about 1 minute, until the ingredients are just combined.
We would also suggest using a recipe which contains part butter and part cream cheese. This is usually a bit more stable since the butter adds slightly more fat and is less likely to dissolve the sugar. Nigella has a Buttery Cream Cheese Frosting recipe in Kitchen (p249) which uses equal quantities of butter and cream cheese - it is whizzed together in the food processor but you can make it with a regular electric mixer by briefly beating the butter until softened slightly then beating in the cream cheese until just combined. Finally beat in the icing sugar a little at a time.