Can my data be white noise if the mean >0? According to the auto-correlation method, my time-series is white noise (i.e. 95% of ACF within ±2/√T), yet the data are counts and thus the mean >0.
Are these two facts incompatible?
I'm using the fpp package in R. Here are my data:
library(dplyr)
library(fpp)

rawdata <- c(414,334,439,385,341,338,365,330,403,321,352,339,270,410,372,332,368,377,392,452,410,411,332,329,422,373,457,406,395,510,412,395,472,429,436,342,427,358,372,393,465,422,481,396,374,393,375,366,313,384,294,311)

#and the plot code:

rawdata%>% ts(frequency=13) %>%
ggAcf()

 A: It's almost pointless to talk about white noise in relation to such short time series. Think of this: you have to establish spectral uniformity of the series. The fidelity and bandwidth of spectral decomposition is so low that you can't reliably claim much on this series in terms of whiteness of the noise, in my opinion.
On the second point, the mean being not zero, the answer could be YES to a reformulated question: can the noise in my series be white if the mean of the series is greater than zero? If you have series $x_t=c+\varepsilon_t$, where $c >0$ is a constant, then $E[x_t]>0$ even when $\varepsilon_t$ is white noise. If you remove the bias in your series, and they become zero mean and colorless, thwn why not call them white noise with bias?
A: Your question might have been entitled in the reverse "is there a useful model for my data or is it without significant predictable structure other than the mean "
The distribution of the observed series IS OF NO CONCERN . The distribution of the residuals from a useful model IS OF CONCERN as that is where all the assumptions reside (are placed !).
Your original data is far from white noise with an Actual/Fit and Forecast graph here showing strong/systematic impact for a few periods of the year and a very significant seasonal auto-regressive structure and a significant level shift down at period 43(44) (FOLLOW THE BLUE LINE IN THE FORECAST REGION ) .
The forecasts are a working image of the model ... 
The model is here  and in more detail here 
The residuals from the model  have the following ACF   suggesting "whiteness" i.e. no anomalies , no auto-correlation in the residuals.
Finally the Actuals/cleansed plot is informative as to the latent identified deterministic structure 
Finally your statement about the acf of the original series suggesting "whiteness" is due to the downwards bias introduced by not treating the pulses and the level shift. See Detecting outliers in a time-series for more on this. Additionally models need to detect anomalies since if untreated they inflate the variance of the errors causing incorrect acceptance of the hypothesis of randomness. Prof. J.K.Ord has referred to this as "the Alice in wonderland effect". The problem is that you can't catch an outlier without a model (at least a mild one) for your data. Else how would you know that a point violated that model? In fact, the process of growing understanding and finding and examining outliers must be iterative. This isn't a new thought. Bacon, writing in Novum Organum about 400 years ago said: "Errors of Nature, Sports and Monsters correct the understanding in regard to ordinary things, and reveal general forms. For whoever knows the ways of Nature will more easily notice her deviations; and, on the other hand, whoever knows her deviations will more accurately describe her ways."
A: White noise is defined as intendent with mean equal to zero, so with non-zero mean it is obviously inconsistent with the definition. Time-series can be uncorrelated and have any mean, lack of correlation does not imply anything about the mean.
