Keyword spotting and in particular Wake-Up-Word (WUW) detection is a very important task for voice assistants. A very common issue of voice assistants is that they get easily activated by background noise like music, TV or background speech that accidentally triggers the device. In this paper, we propose a Speech Enhancement (SE) model adapted to the task of WUW detection that aims at increasing the recognition rate and reducing the false alarms in the presence of these types of noises. The SE model is a fully-convolutional denoising autoencoder at waveform level and is trained using a log-Mel spectrogram and waveform reconstruction losses together with the BCE loss of a simple WUW classification network. A new database has been purposely prepared for the task of recognizing the WUW in challenging conditions containing negative samples that are very phonetically similar to the keyword. The database is extended with public databases and an exhaustive data augmentation to simulate different noises and environments. The results obtained by concatenating the SE with a simple and state-of-the-art WUW detectors show that the SE does not have a negative impact on the recognition rate in quiet environments while increasing the performance in the presence of noise, especially when the SE and WUW detector are trained jointly end-to-end.