After sending coins in your Litecoin-Qt client you should leave your client open until there is one confirmation.
When you send coins, your client will send an initial broadcast to peers. If the transaction does not receive a confirmation your client will send subsequent re-broadcasts. The subsequent re-broadcasts will happen within an hour of the initial broadcast and continue indefinitely until the transaction gets a confirmation. Without leaving your client open, these re-broadcasts will not occur.
If other nodes have just started up (e.g. no blockchain yet) or have flushed your transaction due to it not receiving a confirmation after an extended period of time, only the re-broadcast will get the transaction back into that node’s memory pool.
Once a transaction has at least one confirmation though, there is no further need to leave your client open. There are some corner cases where that’s not true (e.g. a block reorg occurs and that transaction confirmation no longer is valid, etc.) but in almost every instance once a transaction has a confirmation it then no longer needs to be re-broadcast.