Although regulatory improvements for air quality in the European Union have been made, air pollution is still a pressing problem and, its impact on health, both mortality and morbidity, is a topic of intense research nowadays. The main goal of this work is to assess the impact of the exposure to air pollutants on the number of daily hospital admissions due to respiratory causes in 58 spatial locations of Portugal mainland, during the period 2005-2017. To this end, INteger Generalised AutoRegressive Conditional Heteroskedastic (INGARCH)-based models are extensively used. This family of models has proven to be very useful in the analysis of serially dependent count data. Such models include information on the past history of the time series, as well as the effect of external covariates. In particular, daily hospitalisation counts, air quality and temperature data are endowed within INGARCH models of optimal orders, where the automatic inclusion of the most significant covariates is carried out through a new block-forward procedure. The INGARCH approach is adequate to model the outcome variable (respiratory hospital admissions) and the covariates, which advocates for the use of count time series approaches in this setting. Results show that the past history of the count process carries very relevant information and that temperature is the most determinant covariate, among the analysed, for daily hospital respiratory admissions. It is important to stress that, despite the small variability explained by air quality, all models include on average, approximately two air pollutants covariates besides temperature. Further analysis shows that the one-step-ahead forecasts distributions are well separated into two clusters: one cluster includes locations exclusively in the Lisbon area (exhibiting higher number of one-step-ahead hospital admissions forecasts), while the other contains the remaining locations. This results highlights that special attention must be given to air quality in Lisbon metropolitan area in order to decrease the number of hospital admissions.