Should I correct their spelling when they are just beginning to write?

When children are just beginning to read and write, they can come out with the most adorable spellings:

Mummys e z de eg (easter egg)
livin rom
cichn
jragon
sdik
luv

Usually these bear some relationship to the word the child is aiming at. Often they have the sounds of the word (phonology) but not the spelling conventions (orthography). For example, in "livin rom" and "cichn" all of the sounds in the words CAN spell those sounds but don't usually in those particular contexts. So C can sound like /k/ or /s/ but usually sounds like /s/ before an I. CH is a very commmon spelling of the /ch/ sound but just happens not to be how it's spelled in the word "kitchen".

Some of the spellings are more guesses at how to spell a particular sound. So, although some children will pronounce "living" as "livin'", many more will spell it that way - the /ng/ and /n/ sounds are really close and it's easy, if you can't quite hear the difference, or can't quite work out what the difference is really, to go "oh, whatever" and put down the one you know how to spell.

Other spellings represent children's concepts of the sounds of words that are actually, in some cases, more accurate than adults'. The word "stick", most adults will tell you, has a /t/ sound after the /s/. In fact, it's not a /t/ but is a sound somewhere between a /d/ and a /t/ - it's not found on its own in English, though if you are a Hindi speaker you will have this as a separate sound. In English it only exists after /s/. So children who spell "stick" as SDIK are actually very cleverly working out what the second sound is in the word. Likewise "dragon" doesn't begin with the same /d/ as in, say, "dam" but in a slightly different sound, somewhere towards /j/. In fact, when you ask children who can't quite read or write yet about the sounds of words sometimes they are more right than adults.

So, if your child makes this kind of mistake in their spelling, should you correct them? Probably not. They will probably be learning phonic patterns for simple spellings in school, and each pattern they learn will help them to get another set of spellings right - so, when they have learned that there are lots of words ending in -TCH but very few beginning TCH-, they will start spelling words with /ch/ in the middle or at the end with TCH and those with /ch/ at the beginning with CH. But the evidence is that just inventing the spellings themselves can help them with sounding out words and working out what sounds there are in words - essential skills for beginning reading and spelling. Personally, I'd just leave them to it, and have a quick giggle or an "awww" at their productions.

If you're interested in reading more, here's a representative article:

Joseph K. Torgesen, Charlotte Davis, Individual Difference Variables That Predict Response to Training in Phonological Awareness, Journal of Experimental Child Psychology, Volume 63, Issue 1, October 1996, Pages 1-21