I realise this is an old post but perhaps this answer will be useful for others.
Firstly, reinforcement learning is based on the idea of searching for the best long term reward. That is why, in a Q learning algorithm, we update the Q values (or 'goodness' values') for each state-action pair to be equal to the reward received plus some fraction (the rate of decay/gamma) of the predicted future reward. In this way, your algorithm could be converging on good Q values that are considering both expected immediate reward plus potential future rwards.
That being said, if your neural network is indeed diverging, then there are a number of things that you can do to help your algorithm converge. My immediate advice would be to use Double Deep Q learning, whereby you introduce a second neural network and copy the weights from your current neural network every so often (less often than the current network is updated) and use this to provide value predictions for the future state.
So for a neural network that takes a state (your input values) and outputs a range of values inside a list (the indices of which correspond to different actions). This is how you would get each new input, target pair to train your model on:
action = action_the_agent_did_in_this_memory_from_state_to_new_state
target = model.predict(state) #this gives a list of values for each action in the current state
future_target_one = model.predict(new_state) #this gives a list of values for each action in the next state as predicted by your current model
future_target_two = target_model.predict(next_state) #this gives a list of values for each action in the next state as predicted by your target model
best_future_action_index = np.argmax(future_target_one) #this gives the index of the maximum value action in the future state using the current model
best_future_action_value = future_target_two[best_future_action_index] #this sets best_future_action_value equal to the value from the target model (using the index from the current model)
#if this is the last move before the game ends, then there is no future reward
targets[action] = reward #the value of targets at the index equal to the action (such as action 0 perhaps) is set equal to the reward
else: #otherwise one must consider the future rewards too
targets[action] = reward_t + GAMMA * best_future_action_value
This idea is used to decouple the index and value from each other in the value predictions to help prevent problems with overestimation. I hope this helps and you weren't put off by my super-long variable name.