Bleeding mixed with discharge during sex, especially with a new partner, usually points to an infection of the cervix or vagina, such as an STI, rather than an injury. Cervical infections often bleed easily on contact without causing much pain, which fits your pattern well.
Next Steps
A pelvic examination with a vaginal swab and STI screening, including for chlamydia and gonorrhoea, will help find the exact cause. It's best to avoid intercourse until you're examined, so the infection doesn't spread further or get missed.
Health Tips
Please see a gynaecologist soon for the exam and tests. Go in sooner if the bleeding gets heavier or you develop pelvic pain or fever.