Physicians often face tight resource constraints, meaning they have to make trade-offs between which patients they care for and the amount of care received. Studies show that patients requiring many resources disproportionately suffer a loss of care when resources are constrained. This study uncovers whether physicians' attitudes toward prioritization of healthcare predicts poor-health patients' access to care. We combine unique survey data on Danish GPs' preferred prioritization principle with register data on their patients' contacts in general practice. We consider different types of contacts as the required effort could impact the need for prioritization. Our results show variation in GPs' prioritization principles, where a majority prefers a principle that may lead to an unequal distribution of services. We further find that GPs' attitudes toward prioritization predict some poor-health patients' access to general practice. GPs who state they prefer the principle of prioritizing patients in the poorest health state when resources tightened provide more contacts to poor-health patients. The additional contacts are typically high-effort contacts such as annual status meetings and home visits, but also low-effort contacts such as emails. Our findings indicate inequity in poor-health patients' access to care across general practices.