Although various deep keyword spotting (KWS) systems have demonstrated promising performance under relatively noiseless environments, accurate keyword detection in the presence of strong noise remains challenging. Room acoustics and noise conditions can be highly diverse, leading to drastic performance degradation if not handled carefully. In this paper, we propose a noise management front-end called SE-SPP Net performing speech enhancement (SE) and speech presence probability (SPP) estimation jointly for robust KWS in noise. The SE-SPP Net estimates both the denoised Mel spectrogram and the position of the speech utterance in the noisy signal, where the latter is estimated as the probability of a particular time-frequency bin containing speech. Further, it comes at relatively no cost in model size when compared to a model estimating the denoised speech. Our SE-SPP Net can improve noisy KWS performance by up to 7% compared to a similar sized state-of-the-art model at SNR -10dB.