This is possibly the most frustrating stage of learning the game of chess, for you as well as them. Your players know the moves, which are the basic building blocks of the game, but they are not yet able to construct anything with them. It does not seem to matter how old a player is, there are plenty of secondary aged pupils who get stuck at this stage and it seems to me that the best remedy is lots of practice.
I remember some years ago watching two weak players in a tournament in Stowmarket. I was standing alongside Bob Jones, who is probably the most effective chess organiser I have ever met. In the late 1980s and early 1990s, Suffolk produced some tremedous players and won or came close to winning National Titles which should have been out of the question for a rural county with a small population and a lot of this was due to Bob. (Nick Pert, from Ipswich, went on to win the World under 18 Championship in 1998).
The weak players in question had been turning up to tournaments for several months, but clearly learning little. I was mildly surpised by Bob's reaction to the appalling chess going on in front of us: he was on the point of tearing out his hair, and it was an extremely frustrating experience for a reasonably strong player to witness such chess. I was fascinated and I was trying to force myself to understand why it was that these players had not progressed beyond an extremely basic level of play. I still don't think that I have understood this, and I am not sure that I ever will, but this incident demonstrated one thing very clearly: these children were enjoying themselves, illegal moves and all, and it was nobody's business but their own. It is not our job as teachers to interfere during a game because that will kill the child's enthusiasm. Having said that, it is our job to try to help them to play better chess and I think that for this development to happen our players need plenty of practice.
There are others who quickly get to grips with the basics and stop blundering their pieces and moving their kings into check. These players are on your side and it may well be that they will prove better at conveying some of the ideas to the weakest better than you do yourself.
The reputations of chess coaches is never made with the weak players, but I believe that we should try to keep them in the game as long as they are enjoying it. There will be some improvement, and lots of easy exercises to establish good habits will help. If we are using Chess as a vehicle to improve academic performance, it well may be this group which is likely to benefit the most.
In this section, I have tried to keep the emphasis on the King, because in my experience nothing causes to much trouble to the learning player.
It is at this stage that a school competition can be introduced. Firstly, there is the "Ladder". This can take a number of forms. Dedicate a notice board, pin slotted card on, number each slot from 1 to n, write the players' names on card inserts and put them in the slots, randomly at first. Any player may challenge anyone higher than him or herself to a game and a declined challenge is considered to be a loss. If the lower player wins, the two cards swap places. I saw a very neat variation on this in a Kent primary school (Amherst, Sevenoaks, I think) in which a roll-up vinyl chess board had been sacrificed and cut into two so that the long diagonal now formed the base of a triangle of 36 squares. This was glued to plywood and hung on a wall so that the board's corner square now formed the apex. Cup hooks were screwed in and metal "dog tags" each had a sticky lable bearing a child's name. This actually represents beautifully the chess playing community: in a school chess club of 36 players, there is one "King of the Castle" and probably only a couple of players who can realistically hope to challenge him. Those two, in turn might be challenged by half a dozen others, but there are more who might occasionally beat them. A ladder competition can be operated very well by the children themselves, with perhaps some small prizes each half term for improvers and the players who have spent most time at the top.
A more structured approach can also be adopted and for this the players are divided into Leagues of 4 with promotion and relegation every 6 games (they play each other twice, once with white and once with black). If you decide to allow chess to be played every lunchtime, you might have three days' Ladder, one day Leagues and then one day Puzzles (perhaps this humble Webmaster's work might prove useful here) or, if your computer bank is near the chess area, then a rota can be established to decide whose turn it is on the Website today...