Coalition government can possibly have a greater influence on the position of prime minister than any other factor or event. As prime ministers have always had to be mindful about what could pass in the Commons, prime ministers in single party government had important knob to maintain the party in check (patronage powers, institutional (agenda setting) powers, information powers). In a coalition government there is a more formal need to get the approval of the other party.
The experience of coalition government has strengthened the prime ministerial power as it has given the prime minister more power from both parties. The prime minister has a minority party that he can run as well as his own party, which makes him have an advantage of making any amendments, he will have the approval if his party ministers and the Liberal democrats ministers. This increases the level of legitimacy because it enables the coalition to have simple agreements with no disagreements (often some disagreements occurs). An example of this is the pupil premium. This is when support is given to some schools that are in dept. This support is given by the Conservatives which appeals to the Liberal democrats and they support it.
Another similar factor that has strengthened the prime ministerial power is that once policies have been agreed by the coalition partners, they have additional authority because they are supported by two parties and so represent a consensus view. The coalition also has a strong mandate as the two parties together were elected by a majority of the electorate. This is effective as because they have a strong mandate, it will be unproblematic for them to win the next general election. Furthermore, the constant threat of government collapsing has, arguably, strengthened the ability of the prime minister has maintained the coalition discipline.
Another factor that has strengthened the prime ministerial power is that due to the fact that the Conservative party lack a majority, they have to compromise and have to be more central in order for the UKIP to be stronger. This can make less cover on immigration and European union. Also, the coalition has given the prime minister and his government a solid majority in the House of Commons. The government has not lost a single vote in the Commons since the Coalition which is a great success. This is significant as having a coalition for the first time in a long time has gained some useful authority within the government.
Another factor that has strengthened the prime ministerial power is that the prime ministers makes a number of promotions and demotions at once, substantially changing the composition of their cabinet, which is referred to as a 'cabinet reshuffle'. this gives prime ministers a real edge because they can surround themselves with like minded allies who share the same policy aim. for example Thatcher replaced the 'wets' with the 'dries' who shared her vision. David Cameron has also appointed a member of fellow modernising conservatives to his cabinet. Due to the fused executive and legislative branch, these reshuffles can be a good opportunity to reward the loyal backbench MPs with promotion, demote unpopular or 'failing' ministers. It's a good way to keep control.
On the other hand, some people may argue that the coalition government has weakened the prime ministerial power as because of the coalition agreement between the two parties, key appointments have to be agreed with Nick Clegg. This is an issue as every time he makes a policy he would need Clegg's consultant, and he may not agree with everything that Cameron claims. For example, Clegg had to agree with the pupil premium before it could be taken into action. To conclude this, the coalition government has weakened the prime ministerial powers because David Cameron's powers of patronage are limited because he cannot appoint all members himself, Clegg will also be involved with this process. He has to share his powers with Clegg and this makes his powers weakened. Also, policy has to be cleared with his coalition partners and this has caused problems.