It's a complicated, tough call. Here's my situation, which will hopefully be somewhat informative and useful. I've been working as a journalist (mostly opinion writing rather than reporting) since I graduated in '06, and have realized that the issues I care the most about all connect to poverty, inequality, and the wealth gap. I decided to take a crack at grad school with an eye toward becoming a much more informed, connected-to-policy-debates journalist. So I applied to HKS, WWS, and Goldman, and got into the latter two. (It looks like I'll be going to WWS.)
I definitely had less policy background than almost anyone in my incoming class, and certainly couldn't say specifically what I wanted to study, but I made a conscious effort to tell a coherent, well-put-together story in my personal statements: I no longer want to just be another pundit who chimes in constantly and superficially; I want to actually contribute to the discussion about inequality in a careful, well-informed way. I made it clear that my interests were broad and I wanted to narrow them down, and listed some possible areas of focus (education, health care, criminal justice). I also talked about how lucky and privileged I have been growing up, and how I've come to realize just how much was given to me -- all of which is true, but also included to help make it clear to the adcom folks just how visceral an issue the wealth gap has become to me.
As for the WWS policy memo, I made sure that my topic tied closely to inequality, and that every step of the way I evaluated the nuance and complexity of the issues I was discussing (don't quote me on this, but while the point of the policy memo is to suggest concrete solutions, I really do get the sense that they'd rather see a memo filled with caveats and "buts" and "on the other hands" than one that is too sure of itself -- it's not as though they expect you to actually solve the problem in question, so I think it's more of a matter of them getting a gauge for your talents as an analytical thinker who can evaluate various bits of potentially conflicting information).
Oh, and I also told the people writing my letters what I wanted to focus on, which you should definitely do if you apply -- it helps bring the whole application package together.
I feel very thrilled and lucky to have gotten into WWS (anyone who tells you there isn't luck involved is lying -- just look at the numbers), and I was able to do so without a hugely specific area of interest. A few caveats: I write for a living, so I was probably able to make up for my (numerous) other shortcomings as an applicant with a well-written policy memo and personal statement. I also had a very high GRE score, and happen to work for a publication most people have heard of. Plus I have heard that MPP/MPA folks like journalists. I also included a note about my weaknesses (some low early-college grades and a lack of econ/stats course experience), figuring it would be better to point them out and explain them rather than have the adcom people discuss them without me getting a word in -- no idea if this helped, but in the case of WWS it can't have hurt too badly (Goldman didn't have space for such a document, if memory serves, so I only submitted it to WWS and HKS).
But while all the writing/GRE/career stuff might have helped, I do think what worked the most in my favor was the fact that I was able to give the schools a rather strong sense of who I am and what is driving me to study public policy.
So in short, if you can put together an otherwise compelling application that can at least indicate some broad area of interest, I think you'll be in good shape. But I'd ask around and even email some admissions personnel (if you're paranoid such emails would hinder your chances -- though I don't think they would -- just send these emails to competitive programs to which you won't be applying).
Hope this has been helpful, and let me know if you have further questions. I'm by no means an expert, but I did learn a lot from the application process.