LYNN – As Gov. Deval Patrick explores whether the state should house a wave of children illegally crossing into the country, local health care workers Thursday said their resources are already limited by the existing, and growing, immigrant population.”It’s a whole unpaid-for population that is getting care, having babies, having depression, getting cancer, and who’s paying for that stuff?” Community health nurse and Lynn resident Jennifer Beckwith said. “Everyone should have access to health care, but who’s going to pay for it is my worry as a concerned taxpayer.”Unaccompanied children fleeing violence in Central America have been arriving at the southwestern border of the United States by the thousands.By Sept. 30, 90,000 children are expected to arrive in the country.President Obama has requested $3.7 billion in an emergency spending request to help address the situation at the border. Republicans have been pushing to significantly pare down that request.Patrick on Wednesday said he was looking into whether it would be feasible to offer shelter to some of the more than 50,000 children who have already crossed the southwestern border.A recent report from the Massachusetts Office of Refugees and Immigrants shows that of the 725 projected new refugee arrivals, 28 percent are expected to end up in Lynn.But Lynn Mayor Judith Flanagan Kennedy indicated this week that compassion fatigue is setting in.”Lynn has always been a city of immigrants, it’s always been welcoming,” Kennedy told The Daily Item. “We understand the need, but there comes a time when enough is enough.”Kennedy and Lynn Superintendent of Schools Catherine Latham cited an already overcrowded school system where “out of country” admissions increased from 36 for the 2010-2011 school year to 200 in 2011-2012 and 538 new out-of-country admissions in 2013-2014.But new students don’t just need a desk, they need to be immunized, too.City Health Director MaryAnn O’Connor said immunizations have increased by 200 percent over the last couple of years. She said that for many children – whether they are unaccompanied or part of a family? there is no official immunization record, and the child has to receive five shots.O’Connor said the State Department of Public Health supplies the vaccinations at no charge.But the overall influx of immigrants to Lynn – whether entering the country legally or illegally, whether coming as a lone child or as part of a multi-generational family, whether from Asia or the Americas – has definitely affected the health department’s budget.”It’s not just unaccompanied minors; it’s the influx of unaccompanied minors in addition to other immigrants and refugees,” O’Connor said. “It’s impacted the amount of service we’ve had to provide.”Within the last year, the department has hired a translator and a nurse to help staff clinic hours held twice a week.”We’re so lucky that the part-time nurse we hired is also a Spanish-speaking nurse,” O’Connor said. “We need to do paperwork, need to explain to parents (what is happening) … and then need to calm down the child who’s getting five shots.”Any cases of tuberculosis can be particularly labor-intensive. Public health nurses have to witness a patient take daily medication for tuberculosis – and treatment can last for up to nine months, O’Connor said.She said the state pays for the medication, but the city pays for the nursing.Meanwhile, the department has coordinated with the New American Center and the city Office of Emergency Management to try and ensure immigrants have safe housing, a problem O’Connor said needed to be better addressed, and that new arrivals know what to do in emergency situations.Beckwith travels to these homes to visit patients in Lynn, Chelsea, Revere and East Boston. She treats only adults, so she said her resources would not be devoted to any of the unaccompanied minors from Central America, should they arrive.However, Beckwith said her job has changed in the past 10 years from predomi